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Efficient tomography of a quantum many-body system: Supplementary Material

B. P. Lanyon,1, 2, ∗ C. Maier,1, 2, ∗ M. Holzäpfel,3 T. Baumgratz,3, 4, 5 C. Hempel,2, 6 P. Jurcevic,1, 2 I.Dhand,3 A. S. Buyskikh,7 A. J. Daley,7 M. Cramer,3, 8 M. B. Plenio,3 R. Blatt,1, 2 and C. F. Roos1, 2

1Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck, Austria

2 Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria3Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany

4Clarendon Laboratory, Department of Physics,University of Oxford, Oxford OX1 3PU, United Kingdom

5Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom6ARC Centre for Engineered Quantum Systems, School of Physics,

The University of Sydney, Sydney, New South Wales 2006, Australia.7Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK8Institut für Theoretische Physik, Leibniz Universität Hannover, Hannover, Germany

(Dated: July 7, 2017)

Contents

I. Trapped-ion quantum simulator 3A. Multi- and single-qubit rotations 3B. Preparing the Néel state 3

II. Modelling the simulator 3A. Ideal simulator model 3B. Interaction range in experiments 4

III. Measuring and reconstructing local reductions 4A. Chosen measurement setting 4B. Using measurement outcomes 5

IV. Certified MPS tomography 6A. Details of procedure 6

1. Measurements 72. Uncertified MPS tomography 73. Fidelity lower bound and selection of a parent Hamiltonian 84. Statistical analysis of the fidelity lower bound 95. Estimator for the parent Hamiltonian energy 116. Proof for basic variance relations 13

B. Simulations of MPS tomography and certification 14C. Resource cost for a constant estimation error in MPS tomography. 15D. Modelling initial Néel state errors for the 14-spin experiments 16

V. Extended experimental analysis and results 16A. Single site magnetisation dynamics for 8 and 14 spin quenches 16B. Local reductions and correlation matrices for the 8-spin quench 18C. Correlation matrices for the 14-spin quench 21D. Certified MPS reconstructions for 8 spin quench 22E. Reconstructed MPS non-separable across all bipartitions 23

VI. Direct fidelity estimation 23A. Overview of direct fidelity estimation procedure 23B. Mean-square error and bias of DFE estimates 24

VII. Certified MPS tomography is efficient for 1D local quench dynamics 27A. Summary of the results 27B. Background: parent Hamiltonian certificates 28

∗These authors contributed equally to this work.

Efficient tomography of a quantum many-body system: Supplementary Material

B. P. Lanyon,1, 2, ∗ C. Maier,1, 2, ∗ M. Holzäpfel,3 T. Baumgratz,3, 4, 5 C. Hempel,2, 6 P. Jurcevic,1, 2 I.Dhand,3 A. S. Buyskikh,7 A. J. Daley,7 M. Cramer,3, 8 M. B. Plenio,3 R. Blatt,1, 2 and C. F. Roos1, 2

1Institut für Quantenoptik und Quanteninformation,Österreichische Akademie der Wissenschaften, Technikerstr. 21A, 6020 Innsbruck, Austria

2 Institut für Experimentalphysik, Universität Innsbruck, Technikerstr. 25, 6020 Innsbruck, Austria3Institut für Theoretische Physik and IQST, Albert-Einstein-Allee 11, Universität Ulm, 89069 Ulm, Germany

4Clarendon Laboratory, Department of Physics,University of Oxford, Oxford OX1 3PU, United Kingdom

5Department of Physics, University of Warwick, Coventry CV4 7AL, United Kingdom6ARC Centre for Engineered Quantum Systems, School of Physics,

The University of Sydney, Sydney, New South Wales 2006, Australia.7Department of Physics and SUPA, University of Strathclyde, Glasgow G4 0NG, UK8Institut für Theoretische Physik, Leibniz Universität Hannover, Hannover, Germany

(Dated: July 7, 2017)

Contents

I. Trapped-ion quantum simulator 3A. Multi- and single-qubit rotations 3B. Preparing the Néel state 3

II. Modelling the simulator 3A. Ideal simulator model 3B. Interaction range in experiments 4

III. Measuring and reconstructing local reductions 4A. Chosen measurement setting 4B. Using measurement outcomes 5

IV. Certified MPS tomography 6A. Details of procedure 6

1. Measurements 72. Uncertified MPS tomography 73. Fidelity lower bound and selection of a parent Hamiltonian 84. Statistical analysis of the fidelity lower bound 95. Estimator for the parent Hamiltonian energy 116. Proof for basic variance relations 13

B. Simulations of MPS tomography and certification 14C. Resource cost for a constant estimation error in MPS tomography. 15D. Modelling initial Néel state errors for the 14-spin experiments 16

V. Extended experimental analysis and results 16A. Single site magnetisation dynamics for 8 and 14 spin quenches 16B. Local reductions and correlation matrices for the 8-spin quench 18C. Correlation matrices for the 14-spin quench 21D. Certified MPS reconstructions for 8 spin quench 22E. Reconstructed MPS non-separable across all bipartitions 23

VI. Direct fidelity estimation 23A. Overview of direct fidelity estimation procedure 23B. Mean-square error and bias of DFE estimates 24

VII. Certified MPS tomography is efficient for 1D local quench dynamics 27A. Summary of the results 27B. Background: parent Hamiltonian certificates 28

∗These authors contributed equally to this work.

Efficient tomography of a quantummany-body system

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4244

NATURE PHYSICS | www.nature.com/naturephysics 1

http://dx.doi.org/10.1038/nphys4244

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2

C. Product states have simple parent Hamiltonians 29D. Parent Hamiltonian for a locally time evolved state 29E. Conclusion 32

References 33






3

I. TRAPPED-ION QUANTUM SIMULATOR

A. Multi- and single-qubit rotations

We identify the two electronic Zeeman states |S 1/2,m = +1/2〉 and |D5/2,m′ = +5/2〉 of trapped 40Ca+

ions with the |↓〉 and |↑〉 states of spin-1/2 particles, respectively. These atomic states are coupled by anelectric quadrupole transition at the optical wavelength of 729 nm. The quantum states are coherentlymanipulated using a Ti:Sa CW laser with a linewidth of about 1 Hz, Two laser beam paths are employedfor this. First, a global beam illuminates the ion string approximately equally, from a direction perpen-dicular to the ion string axis and at an angle approximately half way between x and y. Consider thestandard Pauli spin operators on spin j: σ j

x, σ jy and σ j

z . The global beam is used to perform global σx andσy rotations simultaneously on all spins, e.g. Gx(θ) = exp(−iθ

∑Nj=1 σ

jx) and Gy(θ) = exp(−iθ

∑Nj=1 σ

jy).

Second, a single-ion-focused beam comes in parallel to the global beam but from the opposite direction.The direction of this beam can be switched to have its focus pointing at different ions within 12 µs, usingan acousto-optic deflector. This addressed beam is frequency-detuned by about 80 MHz from the spintransition and thereby performs an AC Stark rotation on the addressed ion j given by Rj

z(θ) = exp(−iθσ jz).

The combination of a global resonant beam and a focused detuned beam, inducing AC-Stark shifts forcarrying out arbitrary single-spin rotations, has the advantage of not requiring laser beam paths whoseoptical path length difference is interferometrically stabilized. For an overview of the use of global andaddressed beams to manipulate ionic spins (qubits) see [1].

B. Preparing the Néel state

The Néel state is prepared using a combination of global and addressed pulses. The addressed operationAz(θ) = exp(−iθ

∑[ j=[1,3,5..N−1] σ

jz) is employed, corresponding to (ideally) equal rotations around the z

axis of a subset of spins in the string performed sequentially. To create the Néel state from the initial state|↓〉⊗N = |↓, ↓, ↓ ...〉 requires flipping every second spin to the |↑〉 state. This is done by subjecting the initialstate |↓〉⊗N to the following composite pulse sequence Gx(π/4)Gy(π/4)Az(π/2)Gy(π/4)Gx(π/4) |↓〉⊗N thatmaps the initial state onto the Néel state |↑, ↓, ↑ ...〉. To first order, the state created by this sequence isinsensitive to errors in the rotation angles of the x and y rotations. These errors result from the unequalcoupling strength of the global beam across the string, due to its Gaussian transverse intensity profile.We prepare the Néel state for 8 and 14 ion strings. The fidelities of these states with the ideal Néel stateis obtained by directly measuring in the z-basis on all spins. For 8 spins, the fidelity with the Néel stateis 0.967 ± 0.006, corresponding to an average error per spin of − log2(0.967)/8 = 0.006 ± 0.001. For14 spins, the fidelity with the Néel state is 0.89 ± 0.01, corresponding to an average error per spin of− log2(0.89)/14 = 0.012 ± 0.001. Clearly the error-per-particle is significantly larger for the 14 spininitial state, than for 8. Note that the aforementioned directly measured fidelities of the initial state agreewell with the certified lower bounds obtained for measurements on single sites, via MPS tomography(see main text). The increase in error-per-particle for the 14-spin initial state is likely due to clipping ofour single-ion-focused laser beam on optics as it tries to access ions at the ends of longer strings, leadingto larger intensity fluctuations. This should be straightforward to overcome in a new optical setup.

II. MODELLING THE SIMULATOR

A. Ideal simulator model

In the main paper we give approximate spin-spin interaction power-law ranges (α values), light-likecones for information spreading (Figures 2a and 4a) and, in this supplementary material, we comparedata with a theoretical model for our simulator. In this section we explain how we do this modelling.

The full Hamiltonian used for ideal modelling of our simulator dynamics is given by an Ising-typemodel with a transverse magnetic field:

HIsing = ∑i< j

Ji j σxiσ

xj +

N∑i=1

(B + Bi)σzi . (1)

The N × N spin-spin coupling matrix Ji j, parametrising this model, depends on the ion string vibrationalmode frequencies, eigenvectors, detuning from the laser fields, ionic mass and laser-ion coupling strength(the dependency of Ji j on the experimental parameters is elaborated in the supplementary material of [2]).All else held constant, the interaction range (modelled by a power law, see section II B) can be changed






4

by a single experimental parameter: the detuning of laser fields from the motional resonances. Weindependently measure all the aforementioned parameters in our experimental system and thereby derivethe spin-spin coupling matrix Ji j. The transverse field consists of an overall constant field B and site-dependent perturbations Bi. These small inhomogeneities in the transverse field act like local potentialbarriers to spin excitations hopping around the string. They come from e.g. electric quadrupole shiftswhich vary along the string, a residual magnetic field gradient and field curvature along direction of thestring (in addition to our standard constant 4 Gauss field), and AC Stark shifts of the spin transitionsdue to the presence of laser fields with intensity gradients across the string (Gaussian beam profiles).After optimizing the experimental apparatus, the inhomogeneities are small (compared to the spin-spincoupling strength) and play little role in obtaining an accurate description of our system dynamics at theevolution times considered in the main paper.

In the limit B |Ji j|, which is upheld in all our experiments, the Ising Hamiltonian (1) is equivalentto an XY model in a transverse field. Including the field inhomogenuities, the effective Hamiltoniandescribing our experiments is given by

HXY=∑i< j

Ji j (σ+i σ−j +σ

−i σ+j ) +

N∑i=1

(B + Bi)σzi , (2)

with σ+i (σ−i ) the spin raising (lowering) operator for spin i.In previous work we showed that the simulator dynamics is well described by this model in the low-

excitation regime, that is, when the initial state contains at most one [2] or two [3] quasiparticle exci-tations. In this work we explore the dynamics of highly excited initial states: the Néel state containsN/2 excitations. Note that the XY Hamiltonian preserves the excitation number throughout dynamicalevolutions (subspaces with different excitation number are decoupled).

Time-evolved model simulator states are calculated by brute force matrix exponentiation for up to8 spins, e.g. |φ(t)〉 = exp(−iHIsingt) |φ(0)〉. For 14 spins, this approach takes hours and hours to run,using the computers that we have readily available. Therefore, it was time efficient for 14 spins to usethe Krylov subspace projection methods (Arnoldi and Lanczos processes) which, in the case of sparseHamiltonians, give a substantial speed up and well controlled error bounds [4].

B. Interaction range in experiments

The realised spin-spin interactions are to a good approximation described by a power law dependencefalling of with distance |i − j|: Ji j ∝ |i − j|−α with decay parameter α. In the experiment there are twoways to tune the interaction range: either by varying the laser detuning from the motional resonances orby bunching up or fanning out the transverse modes in frequency space. Here the detuning is directlychosen whereas the mode-bunching depends on the effective trapping parameters (therewith also on thenumber of ions). Comparing the experimental coupling strengths as a function of the distance with idealpower-law decay shows that not possible to extract an unambiguous decay parameter α by a direct fitin real space [2]. However, an effective value for α can be estimated by fitting the eigenmode spectrum(or quasiparticle dispersion relation) of our system with the eigenmode spectrum for power-law interac-tions [2]. The power-law exponent α yielding the best fit provides an estimate for the interaction range:α = 1.58 (8 spins) and α = 1.27 (14 spins).

III. MEASURING AND RECONSTRUCTING LOCAL REDUCTIONS

In this section, we describe the measurements performed in the experiment and how these measure-ments are employed in the analysis. MPS tomography requires the ability to estimate the local reduceddensity matrices of all blocks of k neighbouring spins. On a linear chain of N spins, there are N − k + 1such blocks. A straightforward method of reconstructing all these blocks requires a total of (N − k + 1)4k

measurements, each performed on one of the N − k + 1 local blocks of k spins. Instead, we perform3k measurements, each on the entire system of N spins and use these measurements to infer the localreductions. In this section, we describe these 3k measurements, show that they suffice and how they areused.

A. Chosen measurement setting

Here we recall the straightforward method of reconstructing the local reduced density matrices. Onecan obtain the density matrix of k spins from the expectation values of a linearly independent set of 4k






5

observables. The set of all k-fold tensor products of the three Pauli operators X = σx, Y = σy, Z = σzand the identity operator 1 provides one such set. For example, for k = 2 spins, the density matrix can beobtained from the following 16 expectation values

〈X1X2〉〈X1Y2〉〈X1Z2〉〈X112〉〈Y1X2〉〈Y1Y2〉〈Y1Z2〉〈Y112〉〈Z1X2〉〈Z1Y2〉〈Z1Z2〉〈Z112〉〈11X2〉〈11Y2〉〈11Z2〉〈1112〉 . (3)

In order to obtain the expectation value of say 〈Z1X2〉, the following measurement would be performed inthe experiment: Spin 1 is measured in the eigenbasis of Z while spin 2 is measured in the eigenbasis ofX. We refer to this measurement setting as [Z, X]; the eigenbasis of the i-th vector element provides themeasurement basis for the i-th spin. The measurement setting [Z, X] has four distinguishable outcomes(resolved on the CCD camera in our experiment). We obtain spin up |↑〉 or spin down |↓〉 in the Z basisfor spin 1 and spin up |↑〉 or spin down |↓〉 in the X basis for spin 2. By repeating the measurement [Z, X]many times, we can estimate the four outcome probabilities p↑↑, p↑↓, p↓↑, p↓↓.

These probabilities can be used not only for extracting the expectation value 〈Z1X2〉, but also forextracting the expectation values 〈Z112〉 and 〈11X2〉. This insight generalises to k ≥ 2 spins and to moregeneral measurement settings, and it enables us to estimate the expectation values of the 4k measurementobservables in Eq. (3) from only 3k measurement settings.

For each local block of k spins, the following 3k measurement settings suffice. Each of the k spinsrequires measurement in the basis of the three Pauli operators, thus leading to 3k measurement settings.For instance, consider k = 2. In this case, the 3k = 9 measurement settings are given by

[X, X] [X, Y] [X, Z][Y, X] [Y, Y] [Y, Z][Z, X] [Z, Y] [Z, Z] (4)

Each of the 3k measurement settings has 2k distinguishable outcomes. In total, we estimate 3k × 2k = 6k

outcome probabilities. Each of the 4k expectation values required for obtaining the local reduced densitymatrices can be estimated from this set of 6k outcome probabilities. Therefore, the set of 6k outcomeprobabilities is sufficient to estimate a density matrix on k spins [21].

The 3k measurement settings described above are to be repeated for each of the N−k+1 local blocks onthe chain. Independent measurements for the local blocks would require (N − k + 1)3k measurement set-tings, where measurements are performed on the local blocks and remaining spins are ignored. However,a more judicious choice can provide the required information with fewer measurement settings.

We choose a set of 3k total measurement settings such that measurements are performed on the entirespin chain rather than just the local blocks. We repeat each of the 3k measurement settings on k spinsalong the chain. Specifically, for each of the 3k measurement settings, we split the system into N/kblocks and replicate the same measurement settings on each of the blocks. For instance, the case of k = 2requires the measurement settings

[X, X, X, X, . . . ] [X, Y, X, Y, . . . ] [X, Z, X, Z, . . . ][Y, X, Y, X, . . . ] [Y, Y, Y, Y, . . . ] [Y, Z, Y, Z, . . . ][Z, X, Z, X, . . . ] [Z, Y, Z, Y, . . . ] [Z, Z,Z, Z, . . . ]. (5)

In our experiment, we set k = 3, i.e., we perform measurements in 33 = 27 settings. Formally, we performmeasurements in the 3k with k = 3 different measurement bases

[A1, . . . , Ak, A1, . . . , Ak, . . . ] : Ai ∈ X, Y, Z, i ∈ 1, . . . , k

(6)

on N spins. Each of the chosen 3k measurement settings has 2N distinguishable outcomes. m = 1000outcomes (which could take any of the 2N unique values) were recorded for each of the 27 settings.

To summarize, we choose 3k measurement settings comprising repetitions of local 3k Pauli measure-ments. This brings the total measurement setting requirement down from (N−k+1)4k local measurementsto 3k measurements on the entire chain.

B. Using measurement outcomes

Here we describe how the outcomes obtained from measurement settings in Eq. (6) are used in thesubsequent analysis. The measurement data are used either (i) to reconstruct the state via certified MPS






6

tomography or (ii) to estimate local reduced density matrices on k spins, for instance to estimate 3-spinentanglement.

The measurement data are input to the certified MPS tomography algorithms (Section IV) after con-verting to one out of the following two forms. The first form is that of (N − k + 1)4k local expectationvalues

〈As+1 ⊗ · · · ⊗ As+k〉 : As+i ∈ 1, X, Y, Z

, (7)

where s ∈ 0, . . . ,N − k indicates the position of the local block and i ∈ 1, . . . , k labels sites within therespective local block. An alternate but equivalent form of the input to certified tomography is that of theoutcome probabilities of the 6k non-identity Pauli measurements performed on each of the N − k+1 localblocks. Formally, the following (N − k + 1)6k local outcome probabilities are estimated:

〈Ps+1,a1,b1 ⊗ · · · ⊗ Ps+k,ak ,bk〉 : ai ∈ X, Y, Z, bi ∈ −1,+1,

s ∈ 0, . . . ,N − k, i ∈ 1, . . . , k, (8)

where Pj,ai,bi = |aibi〉 j 〈aibi| projects spin j onto the eigenvector |aibi〉 of the Pauli operator ai (= X, Yor Z) with eigenvalue bi. The methods for estimating quantities (7) or 8 from the measurement data aredetailed in Section IV.

The second use of the measurement data is to reconstruct local reduced density matrices of k neigh-bouring spins, or in other words, to perform full quantum state tomography of the local blocks. We usemaximum likelihood estimation [5] to obtain density matrix estimates ρk

qst from the local k-spin outcomeprobabilities. Quantities of interest are computed from the density matrix estimates, e.g., the entangle-ment measures in Figures 2c,d and 4b of the main text or Entropy in FIG. S8. Error bars in quantitiesderived from local reconstructions are obtained from standard Monte-Carlo simulations of quantum pro-jection noise.

IV. CERTIFIED MPS TOMOGRAPHY

In this section, we describe our method for certified MPS tomography, which is based on the resultsof Refs. [6] and [7]. We use the modified SVT algorithm from [6] and the scalable maximum likelihoodestimation method for quantum state tomography from [7] to obtain an estimate of the unknown lab statefrom experimental measurement data. Because the experimentally measured observables do not containcomplete information on the unknown state, an additional step is necessary to verify the correctnessof the result. We use the assumption-free lower bound on the fidelity between our estimate and theunknown state in the laboratory from Ref. [6] for this purpose; we call such a lower bound on the fidelitya certificate. We refer to the combined procedure of MPS tomography and certification as certified MPStomography.

A. Details of procedure

We discuss how to reconstruct and certify a pure quantum state on N qubits from measurements whichallow for the reconstruction of all reduced density matrices of k neighbouring qubits. As there are N−k+1contiguous blocks of size k on a linear chain of length N, the total measurement effort scales at mostlinearly with N. However, even a number of measurements settings constant in N is sufficient if suitablychosen measurements on all N qubits are performed. Here, we use the outcomes of q = 3k settingsmeasured in the quantum simulator experiment (Sec. III). Our discussion is formulated for N qubits(spin- 1

2 particles), but it equally applies to N qudits.FIG. S1 provides a schematic overview over the following subsections. We will proceed as follows:

Measurement data is split into two parts; the first part is used for MPS tomography while the secondpart is used for certification (Sec. IV A 1). We apply existing MPS tomography algorithms to obtain aninitial estimate |ψest〉 of the unknown state ρlab in the laboratory (Sec. IV A 2). From the initial estimate,a family of candidates for a so-called parent Hamiltonian is constructed and one of them is selected,denoted by H. The parent Hamiltonian H provides the certificate and the certified estimate |ψk

c〉 of theunknown state ρlab is given by the ground state |ψGS〉 of H (Sec. IV A 3). The general approach to obtainthe measurement uncertainty of the fidelity lower bound is derived (Sec. IV A 4). The fact that localprobabilities have been obtained from global measurements in the experiment complicates obtaining themeasurement uncertainty of the fidelity lower bound; remaining technical details related to this issue arecovered at the end (Sec. IV A 5).






7

FIG. S1: Schematic overview of certified MPS tomography. This scheme illustrates the certified MPS tomographyprocess discussed in Sec. IV A. in Supplementary Material. Measurement data from the unknown state ρlab is splitinto two parts. The first part of the data is used to obtain a certified estimate |ψk

c〉 and a parent Hamiltonian H, whilethe second part is used to obtain the value Fk

c ± ∆Fkc of the fidelity lower bound.

1. Measurements

Our method begins with measurements on the unknown lab state ρlab. We use the data from themeasurements described in Sec. III. The data comprise m = 1000 outcomes for each of the q = 3k

different measurement settings (6) on N qubits. The samples are split into two parts of 500 samples each.The first part is used to obtain an estimate of the unknown lab state, while the second part is used toobtain the certificate, i.e. the lower bound on the fidelity between the unknown lab state and our estimateof the lab state. This splitting is performed to avoid any risk of overestimating fidelity by constructing orselecting a parent Hamiltonian (see below) which is tuned to the particular set of statistical fluctuationsin a single set of measurement data. Future work could study whether one can use measurement data ina more efficient way.

2. Uncertified MPS tomography

We obtain an estimate of the unknown lab state by combining two efficient MPS tomography algo-rithms [8]. We use the modified SVT algorithm from Ref. [6] to obtain a pure state. This pure state isused as start vector for the iterative likelihood maximization scheme over pure states from Ref. [7]. Thecomputation time required for both algorithms scales polynomially with the number of qubits N.

The input for the modified SVT algorithm consists of the local expectation values from Eq. (7)(Sec. III). Alternatively, one can specify the input as estimates of the local reduced states on k neigbouringqubits (the difference is only an operator basis change in Hilbert-Schmidt space). In our implementation,we choose the latter and convert the local outcome probabilities from Eq. (8) into local reduced stateswith linear inversion. Linear inversion uses the Moore–Penrose pseudoinverse of the mapMs given inEq. (36) on page 12.

The input for the scalable maximum likelihood estimation (MLE) scheme consists of the local outcomeprobabilities from Eq. (8). In principle, one could perform MLE with more information than only thelocal outcome probabilities. For example, one could extract all pairwise correlations available from theexisting measurement data and provide them to the MLE algorithm. This is another avenue for furtherwork. The scalable MLE algorithm returns an initial estimate of the unknown lab state, denoted by |ψest〉.

In both methods mentioned above, the pure state is represented as a matrix product state [9] withlimited bond dimension ∆. In some cases, we observe that the fidelity lower bound obtained at theend of the scheme decreases if we allow for a larger bond dimension ∆. Presumably, this is a result ofstatistical noise adding spurious correlations to our state estimate, which is prevented by lowering the






8

bond dimension. For 8 qubits, we use ∆ = 2 for t ≤ 2 ms and ∆ = 4 for all remaining times. For 14qubits, we use ∆ = 16 for all times.

3. Fidelity lower bound and selection of a parent Hamiltonian

In the last subsection, we have obtained the initial estimate |ψest〉 of the unknown state ρlab. At thispoint, we do not know whether |ψest〉 is close to the unknown state ρlab. In this section, we obtain acertified estimate |ψk

c〉 and a lower bound on the fidelity between |ψkc〉 and ρlab. The lower bound is

provided by a parent Hamiltonian H and its ground state |ψGS〉 is the certified estimate, i.e. |ψkc〉 = |ψGS〉.

A parent Hamiltonian of a pure state |ψGS〉 is any Hermitian linear operator H such that |ψGS〉 is theground state of H. Let E0 ≤ E1 ≤ . . . be the eigenvalues of H in ascending order with degenerateeigenvalues repeated according to their multiplicities. In addition, let the ground state be non-degenerate,i.e. E0 < E1. Then, a lower bound to the fidelity between the ground state |ψGS〉 and any other pure ormixed state ρlab is given by [6]

〈ψGS|ρlab|ψGS〉 ≥ 1 − E − E0

E1 − E0(9)

with the energy E = tr(Hρlab) of the unknown state ρlab in terms of the parent Hamiltonian H. Notethat H usually is artificial and unrelated to any energy in the physical system in the laboratory. If H is asum of local terms—that is, terms acting non-trivially only on k neighbouring spins—the measurementsdescribed above suffice to obtain the energy E and the fidelity lower bound. It remains to find such aparent Hamiltonian, given the initial estimate |ψest〉.

In order to find a larger-than-zero fidelity lower bound, we have to find a parent Hamiltonian whichmust satisfy two conditions: (i) the ground state |ψGS〉 must be close to the initial estimate |ψest〉 and(ii) the gap E1 − E0 between the two smallest eigenvalues must be much bigger than the measurementuncertainty about the value of E. If condition (ii) is not satisfied, we will not learn anything about thefidelity. The following qualitative argument illustrates that condition (i) is necessary as well: If the initialestimate |ψest〉 is far away from the lab state ρlab, we do not try to find a useful parent Hamiltonianbecause it would be unlikely to succeed. Therefore, we only consider the case where the fidelity of theinitial estimate |ψest〉 and the lab state ρlab is high: In this case, a high fidelity between the ground state|ψGS〉 of the Hamiltonian and lab state ρlab is possible only if the fidelity of |ψest〉 and |ψGS〉 is high as well.

First, we attempt to construct a parent Hamiltonian whose ground state |ψGS〉 is equal to |ψest〉. If thematrix product state |ψest〉 belongs to the class of injective MPS for a certain number k of neighbouringspins [10, 11], then the operator

H =N−k+1∑

s=1

11,...,s−1 ⊗ Pker(ρs) ⊗ 1s+k,...,N , ρs = tr1,...,s−1,s+k,...,n(|ψest〉〈ψest|) (10)

has |ψest〉 as its unique ground state [10, 11]; Pker(ρs) is the orthogonal projection onto the kernel of thereduced density matrix ρs on the k neighbouring spins from s to s+k−1. The injectivity property impliescertain restrictions on possible combinations between the bond dimension D of |ψest〉 and the number ofneighbours k. However, the initial estimate |ψest〉 generally is not an injective MPS for our given k andEq. (10) provides a parent Hamiltonian with degenerate ground state. This violates condition (ii) andleads to a zero fidelity lower bound.

To mitigate the problem of ground-state degeneracy of H from Eq. (10) we construct a parent Hamil-tonian in a different way. Specifically, we introduce a threshold τ ≥ 0 and obtain candidates for parentHamiltonians from

H =N−k+1∑

s=1

11,...,s−1 ⊗ Pker(Tτ(ρs)) ⊗ 1s+k,...,N (11)

where the thresholding function Tτ replaces eigenvalues of ρs smaller than or equal to some threshold τby zero. Depending on the value of the threshold, the initial estimate |ψest〉 may or may not be a groundstate of H. We construct a set of candidates H1, H2, . . . for parent Hamiltonians by considering allpossible values of τ ≥ 0. We then try to find a compromise between conditions (i) and (ii) from above,|ψest〉 and |ψGS〉 being similar and a large gap, by choosing the operator H which minimizes

cD(|ψest〉 , |ψGS〉) − (E1 − E0) (12)

where c > 0 is some constant and

D(|ψ〉 , |ψ〉) def= ‖ |ψ〉〈ψ| − |ψ〉〈ψ| ‖1/2

=

√1 − | 〈ψ|ψ〉 |2 (13)






9

is the trace distance [12]. We obtain a valid fidelity lower bound for any value of the constant c. However,we may obtain a very small lower bound or a lower bound associated with a large measurement uncer-tainty for some choices of this constant. We have used the value c = 5 and we do not observe significantlyhigher fidelity lower bounds for other values of this constant. The results presented in the main text, inparticular Fig. 3a, show that this approach can provide useful fidelity lower bounds. Modifying Eq. (12)or choosing a more optimal value for c in connection with the discussion in Corollary 5 on page 31 hasthe potential to provide improved fidelity lower bounds.

We use the following numerical tools to compute eigenvalues and eigenvectors of parent Hamiltonians.For 8 qubits, the full spectrum of the parent Hamiltonian candidates is computed with the library functionnumpy.linalg.eigh() from SciPy [13]. For 14 qubits, a DMRG-like iterative MPS ground state searchwith local optimization on two neighbouring qubits [9, Section 6.3] is used to obtain two eigenvectors ofthe smallest eigenvalue(s). We use the functions mineig() and mineig_sum() available from the Pythonlibrary mpnum [14]. The second eigenvector is obtained as ground state of H′ = H + 5 |ψGS〉〈ψGS|. Forboth eigenvectors, the quantity 〈ψ|H2 |ψ〉 − (〈ψ|H |ψ〉)2 is monitored to ensure sufficient convergence ofthe iterative search. It should be noted that, strictly speaking, DMRG-like algorithms only provide upperbounds on smallest eigenvalues, but in practice they have been observed to be very reliable [9]. Theresults from a first low-precision eigenvalue computation with MPS bond dimension 8 is used to selecta parent Hamiltonian. The eigenvalues from a second high-precision eigenvalue computation with MPSbond dimension 24 is used to obtain the certificate.

As the certified estimate |ψkc〉 = |ψGS〉 is the result of an eigenvector computation, the eigenvector

computation determines the maximal bond dimension of |ψGS〉: For 8 qubits, its bond dimension mayreach the maximal value of 16 and for 14 qubits, its bond dimension can be up to 24. The case k = 1is an exception; there, it is easy to see that a non-degenerate ground state must be a product state (bonddimension 1).

4. Statistical analysis of the fidelity lower bound

The fidelity lower bound in Eq. (9) relies on using the parent Hamiltonian and information aboutthe unknown state ρlab to estimate the energy E = tr(Hρlab). The information about the lab state isobtained from a finite number of measurement outcomes distributed according to an unknown probabilitydistribution. This leads to uncertainty in our estimate of the energy E. To determine this uncertainty inE, we construct an estimator ε(D) for E, where D represents the measurement data. The term estimatorrefers to a function that uses values of random variables—in our case, the measurement data D—toobtain an estimate ε(D) of the true value E [15]. In this section, we present an estimator ε(D) for E andwe quantify our uncertainty about the value of E with an estimator Vε(D) for the mean squared error ofε(D).

In order to introduce the estimator ε(D), we have to define the measurement data D. The measurementsettings used in the experiment were given by Eq. (6) (Sec. III). Here, we describe each measurementsetting as one POVM Π j and we collect the POVMs for all measurement settings in the POVM setΠM = Π j : j. We describe the 3k measurement settings from Sec. III with 3k POVMs:

ΠM =Π j : j = ( j1, . . . , jk), ja ∈ X, Y, Z, a ∈ 1, 2, . . . , k

. (14)

As there are exactly 3k different values of the POVM label j, we can equivalently treat j as integerj ∈ 1, . . . , q , q = 3k.

Each POVM has 2N distinguishable outcomes, i.e. 2N POVM elements:

Π j =Π jl : l = (l1, . . . , lN), lc ∈ −1, 1, c ∈ 1, . . . ,N

. (15)

The POVM elements are given by

Π jl = P1, j1,l1 ⊗ . . . ⊗ Pk, jk ,lk ⊗ Pk+1, j1,lk+1 ⊗ . . . ⊗ P2k, jk ,l2k ⊗ . . . , (16)

where Pi, ji′ ,li = | ji′ li〉i 〈 ji′ li| projects spin i onto the eigenvector | ji′ li〉 of the Pauli operator ji′ (= X, Y orZ) with eigenvalue li (i = 1, . . . ,N and i′ = 1, . . . , k). Note that the single-qubit measurement basis, asindicated by j1, . . . jk, repeats after k qubits. These POVM elements describe exactly the measurementsettings mentioned in Eq. (6) (Sec. III). A single measurement of one of the POVMs produces a singleoutcome l = (l1, . . . , lN) (Eq. (15)). We will refer to an outcome from a measurement of Π j as y j = l =(l1, . . . , lN).

For simplicity, we consider the case where each Π j ∈ ΠM has been measured exactly m times. Thisallows us to take one single outcome y j from each Π j ∈ ΠM and store them into a vector x = (y1, . . . , yq).From now on, we will refer to x as a “single outcome” or as a “(single) sample”. The complete dataset of






10

m outcomes from q POVMs is then structured as D = (x1, . . . , xm). A single element xi of the dataset D isdistributed according a probability density p(x). The sampling distribution pm describes the distributionof the complete dataset D and is given by pm(D) = p(x1) . . . p(xm). (The explicit form of p(x) and pm(D)will be provided in Sec. IV A 5.)

Our estimator will be given in terms of a real-valued function f (x) of a single outcome x. To describeits properties, we will need the expectation value (often referred to as population average)

Ep( f ) =∫

f (x)p(x)dx. (17)

An expectation value corresponds to an exact value which we want to obtain but cannot access directlybecause we do not know p(x). We only have access to m samples from p(x), given by D = (x1, . . . , xm)(which is the measurement data from the experiment). Using this data, we define the data expectationvalue (often referred to as sample average)

ED( f ) =1m

m∑i=1

f (xi). (18)

The data expectation value is a quantity which we can compute from the samples we have, and we willuse it to estimate the expectation value. The covariance and data covariance are defined as usual by

Vp( f , g) = Ep( f g) − Ep( f )Ep(g), VD( f , g) = ED( f g) − ED( f )ED(g), (19)

and the variance is given by Vp( f ) = Vp( f , f ). The product f g is the usual point-wise product ( f g)(x) =f (x)g(x). We use the following textbook relations (see Sec. IV A 6 below for a proof).

Lemma 1. For two functions f and g of a random variable, we have

Epm [ED( f )] = Ep( f ), (20)

Vpm [ED( f ),ED(g)] =1mVp( f , g), (21)

Vp( f , g) =m

m − 1Epm [VD( f , g)]. (22)

In the next subsection, we will define a function fε with the property

Ep( fε) = E. (23)

Indeed, fε will be a linear combination of simple functions which count how often certain partial mea-surement outcomes have occured in the experiment. The function ε(D), defined by

ε(D) = ED( fε), (24)

then has the property

Epm (ε(D)) = Ep( fε) = E. (25)

In other words, ε(D) is an unbiased estimator of E. This provides the equality

Epm [(ε(D) − E)2] = Vpm (ε(D)), (26)

i.e., the estimator’s mean squared error (left-hand side) equals its variance (right-hand side). We estimatethe estimator’s variance using

Vε(D) =1m

mm − 1

VD( fε). (27)

Combining Eqs. (21) and (22) shows

Epm (Vε(D)) = Vpm (ED( fε)) = Vpm (ε(D)), (28)

i.e., Vε(D) is an unbiased estimator of the variance of the estimator ε(D) under the sampling distributionpm. The definition of fε , which is still missing and rather technical, is given in the next subsection.

We summarize the results from this section. Using Eq. (9), a lower bound to the fidelity between thecertified estimate |ψk

c〉 = |ψGS〉 and the unknown lab state ρlab is obtained:

〈ψkc |ρlab|ψk

c〉 ≥ Fkc ± ∆Fk

c . (29)






11

The value of the fidelity lower bound Fkc and its measurement uncertainty ∆Fk

c are given by

Fkc = 1 − ε(D) − E0

E1 − E0, ∆Fk

c =

√Vε(D)

E1 − E0, (30)

with ε(D) and Vε(D) from Eqs. (24) and (27) and the function fε(x) given in the next subsection. Valuesof the fidelity lower bound Fk

c are mentioned in the main text and shown in Fig. 3a of the main text.The estimator ε(D) will be seen to be a weighted sum of functions of many independent random vari-

ables, in our case we have 27000 random variables from 27 measurement bases and 1000 measurementsper measurement basis. While not all 27 measurement bases contribute equally to the weighted sum,all 1000 measurements do contribute equally and it is reasonable to expect that ε(D) will be distributedaccording to a normal distribution.

5. Estimator for the parent Hamiltonian energy

In the last section, we have replaced the uncertified initial estimate by a certified estimate, and we haveprovided the functional form of the fidelity lower bound which provides the certificate. In this section, thefunction fε will be defined; it is required to obtain values of the fidelity lower bound and its uncertaintyand it must obey Eq. (23).

While the value of the fidelity lower bound Fkc could be obtained from an easier computation than

presented below, determining its measurement uncertainty ∆Fkc requires the full discussion of this sub-

section. Incorporating the fact that the local probabilities of Eq. (8) (Sec. III) have been obtained from theglobal measurement bases of Eq. (6) complicates the computation of the measurement uncertainty ∆Fk

c .The complication comes from correlations which arise from two probabilities being estimated using thesame set of measurement outcomes. The following minimal example illustrates how these correlationscome about: The observable σ(1)

Z ⊗ σ(2)Z is measured on the two-spin Bell state (|↑↑〉 + |↓↓〉)/

√2 several

times and the outcome probabilities p(i)↑,↓ of a measurement of σ(i)

Z on spin i (i = 1, 2) are estimated by

counting how often each (partial) outcome has occured. If the estimate p(1)↑ is larger than the true value

12 due to statistical fluctuations, then the estimate p(2)

↑ is also be larger than its true value 12 because the

two estimates are equal. In addition, if the estimate p(1)↑ is larger than its true value, the estimate of p(1)

↓is smaller than its true value (this holds even for a product state). The variance estimator Vε(D) from thelast subsection takes all these correlations into account; it can be shown to estimate all covariances of thekind we have just discussed.

We continue with the pending definition of fε . The parent Hamiltonian H from Eq. (11) takes the form

H =N−k+1∑

s=1

11,...,s−1 ⊗ hs ⊗ 1s+k,...,N , (31)

where each term hs acts only on k out of the total N qubits. We need to estimate the energy E, given by

E =N−k+1∑

s=1

tr(hsρs), (32)

where ρs is the reduced state of ρlab on sites s, s+1, . . . , s+k−1. Our first step is providing an expressionfor E in terms of the local probabilities from Eq. (8) (Sec. III). For this purpose, we define a POVM setΠL whose outcome probabilities correspond to the named local probabilities:

ΠL =

Qs : s = 1, . . . ,N − k + 1. (33)

The individual POVMs Qs are given by

Qs =

Qsi : i = (a1, . . . , ak, b1, . . . , bk), ac ∈ X, Y, Z , bc ∈ −1, 1

(34)

with c ∈ 1, . . . , k . Their 6k elements are given by

Qsi = Ps,a1,b1 ⊗ . . . ⊗ Ps+k−1,ak ,bk . (35)

As above, Pc,ai,bi = |aibi〉c 〈aibi| projects spin c onto the eigenvector |aibi〉 of the Pauli operator ai (= X, Yor Z) with eigenvalue bi. It is understood that the elements of Qs act only on the sites s, . . . , s + k − 1 ofthe N-qubit lab state ρlab.






12

We will use the linear map

Ms(ρs) = [tr(Qsiρs)]Qsi∈Qs (36)

which maps a given k-qubit state ρs on the vector of POVM probabilities psi = tr(Qsiρs). The POVMelements of each POVM Qs ∈ ΠL span the complete space of operators. As a consequence, they arecalled informationally complete and the identityMsMs(ρs) = ρs holds; here,Ms is the Moore–Penrosepseudoinverse ofMs. This relation provides

E =N−k+1∑

s=1

tr(hsMs(Ms(ρs))) =N−k+1∑

s=1

∑i

csi psi, (37a)

csi = tr(hsMs(ei)), psi = tr(Qsiρs), (37b)

where ei is the i-th standard basis vector. We have accomplished the goal of expressing the energy E interms of the local probabilities psi.

If we had measurement data for the POVM set ΠL available, we could use simple counting functions

θsi(x) =

1, if ys = i with x = (y1, . . . , yN−k+1)0, otherwise

, (38)

where we have combined single outcomes ys from Qs ∈ ΠL into vectors x = (ys)s (as has been done inthe last subsection for Π j ∈ ΠM). It is simple to see that such counting functions estimate probabilities(see below for an explicit example):

Epm [ED(θsi)] = Ep(θsi) = psi. (39)

In this case, we could obtain the function fε by replacing psi with θsi in Eq. (37a).However, our measurement data is data for the POVM set ΠM defined above in Eqs. (14)–(16) and we

must estimate the local probabilities psi from that data. We need to establish a relation between the twoPOVM sets ΠM and ΠL. Because of the particular choice we made for these two sets, it is easy to findnon-negative coefficients csi, jl′ such that [22]

Qsi =∑

jl′csi, jl′Π

(s)jl′ , Qsi ∈ Qs, Qs ∈ ΠL, (40)

where the sum over l′ is over the 2k different partial traces

Π(s)jl′ = tr1,...,s−1,s+1,...,N(Π jl) (41)

of the 2N elements Π jl ∈ Π j. (As before, Π j ∈ ΠM .) The partial traces Π(s)jl′ are rank-1 projectors onto a

particular k-fold tensor product of eigenvectors of Pauli matrices. Therefore, we enumerate them with anindex vector l′ = (l′1, . . . , l

′k), l′a ∈ +1,−1, a ∈ 1, . . . , k where l′a specifies whether the a-th eigenvector

is spin up or spin down in some direction (X, Y or Z).In order to estimate the local probabilities psi from data of the global POVM setΠM , we define counting

functions for occurences of a local part (ls, . . . , ls+k−1) of a global outcome (l1, . . . , lN):

θ jsl′ (x) =

1, if (ls, . . . , ls+k−1) = (l′1, . . . , l′k)

with x = (y1, . . . , yq) and y j = (l1, . . . , lN),0, otherwise.

(42)

To show that θ jsl′ can be used to estimate tr(Π(s)jl′ ρs), we have to complete some definitions. As has been

mentioned above, the sampling distribution pm(D) = p(x1) . . . p(xm) describes the probability distributionof the complete dataset D = (x1, . . . , xm). Single outcomes x = (y1, . . . , yq) contain one outcome y j foreach POVM Π j ∈ ΠM . The single-outcome probability density therefore is p(x) = p1(y1) . . . pq(yq). EachPOVM Π j gives rise to to a discrete probability distribution p( j)

l = tr(Π jlρlab), Π jl ∈ Π j. We embed it intoa probability density via

p j(y j) =∑

l

δ(y j − l) tr(Π jlρlab) (43)

where we have used l as an integer from 1, 2, . . . , 2N . The counting functions defined above then havethe property

Ep(θ jsl′ ) = tr(Π(s)jl′ ρlab) = tr(Π(s)

jl′ ρs). (44)






13

Finally, we can put everything together to obtain the final function fε , which will provide the estimatorε(D) = ED( fε) of E. First, we define

fsi(x) =∑

jl′csi, jl′θ jsl′ (x) (45)

and observe

Ep( fsi) =∑

jl′csi, jl′ tr(Π(s)

jl′ ρs) = tr(Qsiρs) = psi. (46)

We define fε by

fε(x) =∑

s

∑i

csi fsi(x), csi = tr(hsMs(ei)). (47)

Using what we have learned so far, we obtain

Epm (ε(D)) = Epm (ED( fε)) = Ep( fε) =∑

si

csiEp( fsi) = E. (48)

This shows that the function ε(D) is an unbiased estimator of the energy E = tr(Hρlab).This scheme has been implemented as part of the Python library mpnum [14, functionmpnum.povm.MPPovmList.est_lfun_from].

6. Proof for basic variance relations

In this section, we proof three basic equalities used in Sec. IV A 4. Their proof is included for com-pleteness.

Let p(x) be some probability density function, D = (x1, . . . , xm) a dataset of m samples from p, and letpm(D) = p(x1) . . . p(xm) the sampling distribution which describes the probability density of the completedataset D. We will also use the definitions from Eqs. (17)– (19) on page 10. The equalities which weproof here provide relations between expectation values, sampling distribution expectation values anddata expectation values. Above, they have been used to estimate the estimator’s variance from data.Lemma 1 states: For two functions f and g of a random variable, we have

Epm [ED( f )] = Ep( f ),

Vpm [ED( f ),ED(g)] =1mVp( f , g),

Vp( f , g) =m

m − 1Epm [VD( f , g)]

Proof. First equation:

Epm [ED( f )] =∫

1m

m∑i=1

f (xi)p(x1) . . . p(xm)dx1 . . . dxm =mmEp( f ). (49)

For the second and third equation, we first compute

Epm [ED( f ),ED(g)] =∫

1m2

m∑i=1

m∑j=1

f (xi)g(x j)p(x1) . . . p(xm)dx1 . . . dxm

=mm2 Ep( f g) +

m2 − mm2 Ep( f )Ep(g)

=1mVp( f , g) + Ep( f )Ep(g) (50)

This provides

Vpm [ED( f ),ED(g)] = Epm [ED( f ),ED(g)] − Epm [ED( f )]Epm [ED(g)]

=1mVp( f , g) (51)

and

Epm [VD( f , g)] = Ep( f g) − Epm [ED( f ),ED(g)] = (1 − 1m

)Vp( f , g), (52)

which is the required relation.






14

B. Simulations of MPS tomography and certification

In Figure 3a of the main text, fidelity lower bounds Fkc based on an ideal model of the tomographic

process are presented. Here we explain how they were obtained.The ideal model of the tomographic process assumes a perfect initial state |φ(0)〉 = |↑↓↑↓ . . .〉 in the

σz basis and an idealised time evolution of the quantum simulator, described by the Hamiltonian of eq. 1The transverse fields B, Bi and coupling matrix elements Ji j have been described above. Note that inthe limit B |Ji j|, which is upheld in the experiments, the above Hamiltonian is equivalent to an XYmodel in a transverse field, as described in Sec. II. For more details see the supplementary materialof [3]. The ideal time evolution |φ(t)〉 = exp(−iHIsingt) |φ(0)〉 is computed using the library functionscipy.sparse.linalg.expm_multiply [13]. (For the simulation with a mixed initial state discussedin Sec. IV D below, the same library function has been used to propagate 15 different pure initial statesin time. It was convenient to convert the resulting state to a purified MPS representation with a singleancilla site of dimension 15.)

The simulated MPS tomography and certification can be performed with exact knowledge of localprobabilities or data from a finite number of simulated measurements (simulation mentioned in Sec. IV Dbelow).

Exact knowledge of local observables. Assuming exact knowledge of local observables simplifiesthe simulation. The MPS tomography algorithms are run with the exact values of the 6k probabilitiesdescribing the measurement outcomes of the 3k k-fold tensor products of the Pauli X, Y and Z matricesfor each of the N − k + 1 local blocks. Computing the energy E = tr(Hρlab) of the (now known) ideallab state |φ(t)〉 in terms of the parent Hamiltonian H is simplified considerably as we can compute theexact local reductions of ρlab. As a consequence, the resulting fidelity lower bound is known withoutuncertainty as well.

This numerical simulation represents the expected performance of certified MPS tomography in a casewhere the perfect initial state is prepared, the simulator produces ideal unitary dynamics and an infinitenumber of perfect measurements are performed. This is shown in Fig. 3a in the main text.

Finite number of simulated measurements of global observables. While the measurement of aglobal observable such as X⊗N can yield one of 2N outcomes, it is simple to draw a sample from theprobability distribution of measurement outcomes if a matrix product description of the state on whichmeasurements shall be simulated is available: One can simply compute the local reduced state on the firstqubit, simulate a single-qubit measurement there and continue by computing the reduced state state of thesecond qubit conditioned on the outcome of the first measurement, etc. [23]. A more direct way to imple-ment the same procedure uses a matrix product description of the POVM in question to obtain a matrixproduct representation of the measurement outcome probabilities. Marginal and conditional distributionsof the full outcome probability distribution can be efficiently obtained. This procedure has been imple-mented as part of the mpnum library [14] (function mpnum.povm.MPPovm.sample(method=’cond’)).However, with our code for 14 qubits, it was still faster to convert the matrix product representation ofthe outcome probability distribution to a full array with 214 elements and sample from the full descrip-tion. This simulation represents the expected performance of certified MPS tomography on our system,assuming everything is perfect but allowing for a finite number of measurements (1000) per basis. Theoutcome of these simulations are shown in FIG. S2.






15

Time (1/J)

k = 5k = 4

k = 3

k = 2k = 1

FIG. S2: Fidelity lower bounds for the 8-spin quench: comparison with theory. Certified lower bounds on thefidelity between MPS estimates |ψk

c〉, reconstructed from measurements over k sites, and the quantum simulator stateρlab. Shapes: data points with errors, 1 standard deviation. Error bars reflect uncertainty due to finite measurementnumber and can be calculated directly from raw measurement statistics through the certification process (Methods).Data is compared to two theoretical models (dashed lines and shaded areas). Both models consider ideal simulatorstates |φ(t)〉. Dashed lines: using exact knowledge of k-site local reductions derived from |φ(t)〉. Shaded areas: usingimperfect knowledge of k-site local reductions derived from the ideal states by simulating the outcomes of the finitenumber of measurements per measurement basis used in experiments (1000). See Methods. Ones sees that the effectof imperfect knowledge of local reductions accounts for many of the differences between data and exact theory, andthat there isn’t much to gain by taking more measurements. A possible reason for additional discrepancies betweendata and model is mixture in the laboratory state, which will reduce the quality of the pure-state MPS description.

C. Resource cost for a constant estimation error in MPS tomography.

In the final section in the Methods we discuss the resource cost for a constant estimation error inMPS tomography. We performed extensive numerical simulations (see FIG. S3) in order to demonstratethe practicality of the scheme and its efficient scalability to large systems in comparison to full statetomography. We found that for a constant reconstruction fidelity the required number of measurementsscales as the third power in the number of subsystems N. The post-processing time to obtain the stateincreases with ≈ N1.2. Hence MPS tomography involves an experimental and computational effort thatscales polynomially in N while standard tomography scales exponentially.

FIG. S3: Estimation error of MPS tomography with a finite number of measurements. The estimation error Dand estimation fidelity F = |〈ψideal|ψest〉|2 are related by D =

√1 − F. M measurements per basis were simulated

(numerically on a classical computer) on an N-spin ideal state |ψideal〉 and the estimate |ψest〉 was obtained with MPStomography; this procedure was repeated 10 times. Markers show errors obtained in 10 individual reconstructionsand lines connect average values. MPS tomography was applied to a state from a nearest-neighbour quench (seeMethods in the main text). Left: Estimation error as function of number of spins N. Right: Estimation error asfunction of N/

√M; M is the number of measurements per basis. Markers are shifted slightly horizontally to enhance

visibility and straight lines serve as guide to the eye.






16

D. Modelling initial Néel state errors for the 14-spin experiments

In the main text we state that the differences, between the experimentally-obtained and ideal-simulatormodel fidelity bounds F3

c at t14 = 4 ms are largely explained by errors in the initial Néel state preparationfor 14 spins. Here we aim to convince the reader of that.

In section I B of this supplementary material we showed that the initial 14-spin Néel state was preparedwith a (directly-measured) fidelity of 0.89 ± 0.01, compared to 0.967 ± 0.006 for the 8-spin case. Thiscorresponds to a significantly larger error-per-particle (e.g. quantified by log(F)/N) for the 14-spin case:− log(0.89 ± 0.01)/14 = 0.012 ± 0.01, while log(0.967 ± 0.006)/8 = 0.007 ± 0.001

We performed numerical simulations to determine the effect of the errors in the 14-spin initial state onthe MPS reconstruction of the 14-spin quench experiment. Specifically, we asked how detrimental theinitial state errors are expected to be to the performance of the MPS tomography for 3-site measurementsafter 4 ms of evolution (the time at which direct state fidelity estimation was carried out).

To do this, we modelled the initial state error in the following way. Analysis of the direct measurementresults for the 14-spin initial state in the lab show that, out of 1000 times that we prepared and measuredthe state in the z-basis, 893 times we observed the Néel state (hence the fidelity of 0.893). 93 times weobserved a state with one spin flip error. For the remaining 12 times, we observed two spin flip errors.The errors are most likely caused by fluctuations in the intensity of our addressed laser beam, meaningthat these erroneous states are added in mixture with the ideal Neel state. We built a noisy model for theinitial lab state as an appropriately weighted mixture of the ideal Neel state and single spin flip errors.We call this model of the noisy initial 14 spin state ρ14

noisysim.In the next step, we numerically simulate outcomes of the same measurements as have been performed

in the actual experiments in the lab (1000 measurements of each setting described in Sec. III). We insertmeasurement outcomes from numerical simulation into the MPS tomography search algorithm. TheMPS reconstruction then proceeds as usual, as described above. The output is a certified estimate for thefidelity lower bound F3

c,noisy, that we would expect to achieve when measuring such a noisy state in thelab. The result for the 14 spin noisy model and t = 0 ms is the lower bound F3

c,noisysim = 0.90 ± 0.03.This is to be compared with the direct (exact) fidelity measurement of 0.89 ± 0.01 in the lab and theMPS-tomography lower fidelity bound, for measurements on k = 1-site, of 0.90 ± 0.04. Clearly all agreewell and the lower bounds are tight.

The result for the 14 spin noisy model at t14 = 4 ms is F3c,noisysim = 0.49 ± 0.07. This is to be compared

with the lower bound obtained in the experiment F3c = 0.39 ± 0.08. Of course, the described errors in

the initial state are not the only errors in the experiment, however, we conclude that they are largelyresponsible for the difference between the fidelity lower bounds obtained from experimental data andfrom an idealised model of a perfect simulator. So, noise adding mixture to the initial state preparationexplains why the experimentally obtained fidelity lower bounds for 14 spins are lower than the ideal case.In the future, we should work on keeping the error-per-particle constant when scaling up our system.This should be relatively straightforward for up to several tens of spins, by rebuilding the optical setupused to deliver the addressed laser beam. Other sources of error that we considered, and found to playminor roles by equivalent numerical modelling, are: the finite lifetime of the excited spin (atomic) state;correlated dephasing due to correlated fast fluctuations in real magnetic fields around the ion string; smallerrors in the analysis pulses that determine the measurement bases.

V. EXTENDED EXPERIMENTAL ANALYSIS AND RESULTS

A. Single site magnetisation dynamics for 8 and 14 spin quenches

Figures 2a and 4a in the main text present the measured single-site magnetisation dynamics for 8 and14-spin quenches, for the Néel initial state. In order to calibrate our experimental system, and comparethe model dynamics with the results in the lab, we run quenches starting with a spin state that contains asingle spin excitation (local quench [2]). The subsequent dynamics reveal the spreading of correlationsfrom a single site and provide a useful visualisation of the approximate light-like cones (approximatemaximum group velocity for the spread of information). FIG. S4 compares the measured and modelsingle-site magnetisation dynamics for a local quench and the Néel initial state for 8 spins, showing howthe single excitation spreads out in the system. Two kinds of light-like cones are presented, as describedin the caption. The faster of which is the same as those presented in the main text. The maximumrate of information spreading should not depend on the initial state, but only the spin-spin interactionHamiltonian. FIG. S5 presents the same but for 14 spins. The experimental results are the same as thosein figures 2a and 4a in the main text. In all cases, the match between data and model is excellent.






17

Data

2 4 6 8Ion number

0

5

10

15

20

25

30

Tim

e (m

s)

0

1

2

3

4Ti

me

(1/J

)

Theory

Ion number

0

5

10

15

20

25

30

Tim

e (m

s)

2 4 6 8

0

1

2

3

4

Tim

e (1

/J)

Data

2 4 6 8Ion number

0

2

4

6

8

10

Tim

e (m

s)

0

0.2

0.4

0.6

0.8

1

1.2

Tim

e (1

/J)

Theory

Ion number

0

2

4

6

8

10

Tim

e (m

s)

2 4 6 8

0

0.2

0.4

0.6

0.8

1

1.2

Tim

e (1

/J)

-1 1

FIG. S4: Magnetisation dynamics during an 8 spin quench. Time evolution of magnetisation 〈σzi (t)〉 for an initial

single spin excitation (top row) and an initial Néel state (bottom row). Left column: Experimental data. Rightcolumn: predictions from idealised model (|φ(t)〉). The two time axes distinguish between the real time passed inthe laboratory (right axis) and time renormalised by the mean nearest-neighbour spin-spin interaction (Methods:Normalised time units 1/J ). The colours identify the spin state: Dark blue indicates a |↓〉 state, red a |↑〉 state. Inboth data panels, red dashed lines are fits to the peaks of the observed magnetisation (information) wavefronts seen inthe single excitation dynamics (top row). The spreading out (dispersion) of the initially localised spin-excitations canbe approximated bounded by light-like cones: Orange dotted lines show the maximum expected velocity at whichcorrelations spread out, estimated by renormalising the mean nearest-neighbour interaction strength by the algebraictail (see Methods: Light-like cones and interaction ranges). The orange dotted lines are the same for both data panelsand the method to derive them is identical (it does not depend on the initial state). From the top data panel one seesthat the orange light-like cones contain the vast majority of the information wavefronts. The orange dotted linesare the ones shown as black dotted lines in Figure 2 of the main text. The light-like cones presented in this paperserve as a useful guide to interpret correlation spreading in our system and when one should expect measurementsover larger sites to become necessary for successful MPS tomography.






18

Data

2 4 6 8 10 12 14Ion number

0

5

10

15

20

25

30

Tim

e (m

s)

0

0.5

1

1.5

2

2.5Ti

me

(1/J

)

Theory

Ion number

0

5

10

15

20

25

30

Tim

e (m

s)

2 4 6 8 10 12 14

0

0.5

1

1.5

2

2.5

Tim

e (1

/J)

Data

2 4 6 8 10 12 14Ion number

0

2

4

6

8

10

Tim

e (m

s)

0

0.2

0.4

0.6

0.8

Tim

e (1

/J)

Theory

Ion number

0

2

4

6

8

10

Tim

e (m

s)

2 4 6 8 10 12 14

0

0.2

0.4

0.6

0.8

Tim

e (1

/J)

-1 1

FIG. S5: Magnetisation dynamics during an 14 spin quench. Time evolution of magnetisation 〈σzi (t)〉 for initial

single spin excitation (top row) and initial Néel state (bottom row). Left column: Experimental data. Right column:Predictions from idealised model right column (|φ(t)〉). For normalised time units see Methods: Normalised timeunits 1/J. In both data panels, red dashed lines are fits to the peaks of the observed magnetisation wavefronts seenin the single excitation dynamics (top row): the maxima of the leading information wavefronts. The spreadingout (dispersion) of the initially localised spin-excitations can be approximated bounded by light-like cones: Orangedotted lines show the maximum expected velocity at which correlations spread out, estimated by renormalising themean nearest-neighbour interaction strength by the algebraic tail (see Methods: Light-like cones and interactionranges). The Orange dotted lines are the same for both Data panels and the method to derive them is identical (itdoes not depend on the initial state). From the top Data panel one sees that the orange light-like cones containthe vast majority of the information wavefronts. The orange dotted lines are the ones shown as black dottedlines in Figure 4 of the main text. The light-like cones presented in this paper serve as a useful guide to interpretcorrelation spreading in our system and when one should expect measurements over larger sites to become necessaryfor successful MPS tomography.

B. Local reductions and correlation matrices for the 8-spin quench

We now present dynamical properties of the experimentally reconstructed local reductions of singlespins, neighbouring spin pairs and neighbouring spin triplets, during the 8-spin quench experiments.These local reductions are reconstructed directly from the local measurements, using full quantum statetomography (not MPS tomography), which searches over all possible physical states (pure and mixed) tofind an optimum fit with the data.

FIG. S6 presents the dynamics of entanglement, quantified by the logarithmic negativity, in theexperimentally-reconstructed local reductions. Spin pair entanglement maximises at 2 ms and spin tripletentanglement between 3 and 4 ms. As the simulator evolves further, entanglement reduces, first in pairsthen in triplets agreeing with the spread of correlations in the system. The measured results closely fit anideal model of the simulator (not shown for clarity).






19

1 3 5 7Spin pair

0

0.2

0.4

0.6

Ent

ang.

LN

2

Time = 0 ms

1 3 5 7Spin pair

Time = 1 ms

1 3 5 7Spin pair

Time = 2 ms

1 3 5 7Spin pair

Time = 3 ms

1 3 5 7Spin pair

Time = 4 ms

1 3 5 7Spin pair

Time = 5 ms

1 3 5 7Spin pair

Time = 6 ms

1 3 5 7Spin pair

Time = 7 ms

1 3 5Spin triplet

0

0.2

0.4

0.6

Ent

ang.

LN

3

Time = 0 ms

1 3 5Spin triplet

Time = 1 ms

1 3 5Spin triplet

Time = 2 ms

1 3 5Spin triplet

Time = 3 ms

1 3 5Spin triplet

Time = 4 ms

1 3 5Spin triplet

Time = 5 ms

1 3 5Spin triplet

Time = 6 ms

1 3 5Spin triplet

Time = 7 ms

(a) Bipartite logarithmic negativity LN

(b) Tripartite logarithmic negativity LN

2

3

FIG. S6: Evolution of entanglement in local reductions during the 8 spin quench. (a) Bipartite logarithmicnegativity LN2 for neighbouring spin pairs and (b) tripartite logarithmic negativity LN3 for neighbouring spin triplets.The entanglement measure LN3 is defined as the geometric mean of all three bipartite logarithmic negativity splittingsof a triplet (Methods). The simulator evolution time for each panel is shown in its title. The values are extracteddirectly from the experimentally-reconstructed k-spin local reductions (density matrices). Error bars, representing 1standard deviation confidence, are derived with the standard Monte Carlo method to simulate quantum project noise(finite measurement number effects). For clarity, values from an ideal simulator model are not shown: the data islargely indistinguishable from the ideal model for all times.

In Figures 3b,c of the main text, correlation matrices are presented showing correlations in variousbases between spin pairs across the 8-spin system. FIG. S7 presents correlations measured in additionalbases and compares them with those captured in the measured MPS reconstructions and those from anideal model of the simulator.






20

Theory: <YY>-<Y><Y>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

Theory: <XY>-<X><Y>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

Theory: <ZZ>-<Z><Z>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

(a) Ideal theory

Laboratory: <YY>-<Y><Y>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

Laboratory: <XY>-<X><Y>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

Laboratory: <ZZ>-<Z><Z>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

(b) Direct measurements

MPS rec.: <YY>-<Y><Y>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

MPS rec.: <XY>-<X><Y>

2 4 6 8Spin

1

2

3

4

5

6

7

8

Spi

n

MPS rec.: <ZZ>-<Z><Z>

2 4 6 8

1

2

3

4

5

6

7

8

Spi

n

Spin -0.5 0.5

(c) MPS reconstruction

FIG. S7: Two-spin correlation matrices for 8 spins. 〈Yi(t)Y j(t)〉 − 〈Yi(t)〉〈Y j(t)〉 (left column) and 〈Xi(t)Y j(t)〉 −〈Xi(t)〉〈Y j(t)〉 (middle column) and 〈Zi(t)Z j(t)〉−〈Zi(t)〉〈Z j(t)〉 (right column) at t = 3 ms evolution time (see sub paneltitles). Top row (a): Calculated from theoretical ideal state |φ(t)〉. Middle row (b) outcome of direct measurementson the state in the laboratory (requiring more measurements than used for MPS tomography). Bottom row (c):Calculated from the experimentally-reconstructed certified MPS |ψ3

c〉 for the 8 ion Néel state after 3 ms simulatorevolution (using measurements on k = 3 sites). The hatched squares denote correlations that were not measured. Asubset of the above matrices are presented in Figure 3 in the main text. Note: X,Y and Z are the standard Pauli spinoperators.

The Von Neumann entropy (‘quantum entropy’) of example local reductions during the 8 spin quenchare presented in FIG. S8(a). The maximal value for Von Neumann entropy of an N spin (qubit) state isln(2N), corresponding to a maximally mixed state (shown as horizontal dashed lines in the figure).

FIG. S8(b) presents the fidelity between the experimentally reconstructed neighbouring spin-pair statesand a maximally entangled two-spin state (|Ψ+〉 Bell state). The entanglement between neighboursreaches a maximum between 2 and 3 ms. Quantifying entanglement in terms of the fidelity with a maxi-mally entangled two-qubit state has operational meaning: states with fidelities above 50% are distillable.That is, given many copies of states with fidelities above this threshold, fewer states with higher qualityentanglement can be distilled [16].

The fidelities of the directly reconstructed local reductions with the ideal simulator model are pre-sented in FIG. S8(c-e). For fidelity F of an N-spin state, a measure of error-per particle is EN =

log2(F)/N. For the initial states (time t = 0) in FIG. S8(c-e), we find E1, E2 and E3 all agree to withinmeasurement uncertainty. That is, the error in the initial states of single spins, pairs and triplets is statis-tically indistinguishable. The single spin fidelities reach unity after a few ms of evolution. This can be






21

understood by considering that single spin states rapidly become fully mixed due to quantum correlations.The fully mixed state is unchanged under any unitary rotations and many physical noise sources.

0 1 2 3 4 5 6 7Time (ms)

0

0.5

1

1.5

2

2.5

VN

Ent

ropy

3

34

345

0 1 2 3 4 5 6 7Time (ms)

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

Fide

lity

12 ( 78)

23 ( 67)

34 ( 56)

45

0 2 4 6Time (ms)

0.9

0.92

0.94

0.96

0.98

1

Fide

lity

8

0 2 4 6Time (ms)

0.9

0.92

0.94

0.96

0.98

1

Fide

lity

1,2

7,8

0 2 4 6Time (ms)

0.9

0.92

0.94

0.96

0.98

1

Fide

lity

1,2,3

6,7,8

(a) Von Neumann Entropy (b) 2 qubits: fidelity with Bell state

(c) 1 qubit: fidelity with ideal (d) 2 qubits: fidelity with ideal (e) 3 qubits: fidelity with idealFIG. S8: Properties of the local reductions measured during the 8-spin quench. Measurements over k-sitesenables direct reconstruction of the k-site local reductions (density matrices) via standard Maximum Likelihood re-construction. From these experimentally-reconstructed local reductions, various properties can be derived, with errorbars representing one standard deviation obtained via the Monte Carlo simulation of quantum projection noise. (a)Von Neumann entropy in example local reductions of a single spin (blue), two spins (red), three spins (black). Shapes:data, entropy of experimentally-reconstructed local reductions via full QST. Solid lines: model based on ideal quan-tum simulator states. Dashed lines: the maximum entropy for fully mixed state of N spins is N (qubits). Error bars indata are almost smaller than the symbol size. (b) Overlap of the maximally entangled |Ψ+〉 = (|01〉 + |10〉)/

√2 Bell

state with the absolute value of the experimentally reconstructed neighbouring 2-spin density matrices. Spin pairssymmetrically distributed around the centre of the string are shown in the same color. Solid lines connecting pointswith error bars: data. Dashed lines: values from ideal model of the simulator.(c - e) Overlap between the 8-spin state in the laboratory and the ideal state. Time dependent overlap of the measuredreduced single-qubit (c), two-qubit (d) and three-qubit (e) density matrices with the theoretical ideal density matrices.

C. Correlation matrices for the 14-spin quench

In Figure 4c-e of the main text, correlation matrices are presented showing correlations in various basesbetween spin pairs across the 14-spin system. FIG. S9 presents correlations measured in additional basesand compares them with those captured in the measured MPS reconstructions and those from an idealmodel of the simulator.






22

(a) Ideal theory

(b) Direct measurements

(c) MPS reconstruction

Theory: <YY>-<Y><Y>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

Theory: <XY>-<X><Y>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

Theory: <ZZ>-<Z><Z>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

Laboratory: <YY>-<Y><Y>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

Laboratory: <XY>-<X><Y>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

Laboratory: <ZZ>-<Z><Z>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

MPS rec.: <YY>-<Y><Y>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

MPS rec.: <XY>-<X><Y>

2 4 6 8 10 12 14Spin

2

4

6

8

10

12

14

Spi

n

MPS rec.: <ZZ>-<Z><Z>

2 4 6 8 10 12 14

2

4

6

8

10

12

14

Spi

n

Spin -0.5 0.5

FIG. S9: Two-spin correlation matrices for the 14 spin quench. 〈Yi(t)Y j(t)〉 − 〈Yi(t)〉〈Y j(t)〉 (left column),〈Xi(t)Y j(t)〉 − 〈Xi(t)〉〈Y j(t)〉 (middle column) and 〈Zi(t)Z j(t)〉 − 〈Zi(t)〉〈Z j(t)〉 (right column) at time t = 4 ms. Atthis time, the fidelity between the |ψ3

c〉 MPS estimate and lab state is 0.74±0.05 (via direct fidelity estimation). Toprow (a): Calculated from theoretical ideal state |φ(t)〉. Middle row (b) outcome of direct measurements on the statein the laboratory (requiring more measurements than used for MPS tomography). Bottom row (c): Calculated fromthe experimentally-reconstructed certified MPS |ψ3

c〉 for the 14 ion Néel state after 4 ms simulator evolution (usingmeasurements on k = 3 sites). The hatched squares denote correlations that were not measured. A subset of theabove matrices are presented in Figure 4 in the main text. Note: X,Y and Z are the standard Pauli spin operators.

D. Certified MPS reconstructions for 8 spin quench

In Figure 3a of the main text, the fidelity bounds for the 8-spin quench experiment are presented. Inthat figure, the data are compared with a theoretical model (numerical simulation) which shows the MPStomography fidelity bounds that would be obtained for an idealised model of the simulator. Specifically,the exact local reductions of the model state |φ(t)〉 = exp(−iHIsingt) |φ(0)〉 are used as input to the MPStomography algorithm and certification process. This idealised simulation is described in section IV B.We would only expect to achieve these results in the laboratory if first, our system exactly implementedthe XY model Hamiltonian and, second, we carried out an infinite number of perfect measurementsto determine the local reductions. Clearly we do neither of these things. FIG. S2 compares the sameexperimental data with a more realistic model (Shaded areas). This model again uses the ideal simulatorstates but considers the effect of carrying out a 1000 (perfect) measurements per basis to identify localreductions, as done in the experiments. This model is described in more detail in section IV B. From thiswe conclude that: 1. the differences between data and the original perfect model are largely explained






23

by the finite number of measurements used in experiments and: 2. there is not much to be gained fromdoing more measurements in the lab.

E. Reconstructed MPS non-separable across all bipartitions

We calculated the Von Neumann entropies S of the pure reconstructed MPS |ψ3c〉 for all bipartitions

over the string and for each time step, both from experimental 8-spin data and theoretical simulations.

S = −tr(ρ log2 ρ) or also S = −∑

j

η j log2 η j ,

with the reduced state ρ and the eigenvalues η j. We find that the entropy of the reconstructed MPS agreeswith the expected values derived from theoretical simulations. The entropy is observed to increase intime and reaches values between 0.7 < S < 1 for all bipartitions, after 5 ms. This means that—acrossalmost all bipartitions—the same amount of entanglement is present as in a maximally entangled state oftwo qubits (S = 1). The entropies may grow beyond S = 1 at later times, but at those times, we do nothave a positive fidelity lower bound and no certified estimate.

VI. DIRECT FIDELITY ESTIMATION

The fidelity lower bounds returned by the certification procedure described in the main text are merelylower bounds. That is, the actual overlap (fidelity) between the two states could take any value betweenthe certificate and unity. Which value does the fidelity actually take is a natural question to ask. Toestimate the overlap, we implement the method of direct fidelity estimation (DFE) [17, 18]. The DFEmethod uses measurements on a lab state to determine an estimate of the fidelity between the lab state(generally mixed) and a given pure state, which we set as the output MPS from the efficient tomographyprocedure. In this section, we provide an overview of DFE with emphasis on the current experiment.

A. Overview of direct fidelity estimation procedure

Before describing the employed DFE method, we recollect relevant notation. Consider the state ρlab

implemented in the laboratory and let∣∣∣ψ3

c

⟩be the pure-state estimate obtained from MPS tomography on

ρlab. The fidelity of the estimate with respect to the lab state is defined as

F(∣∣∣ψ3

c

⟩, ρlab

)def=⟨ψ3

c

∣∣∣ ρlab∣∣∣ψ3

c

⟩. (53)

Define Pk : k = 1, 2, . . . 4N as the orthonormal Pauli operators tr(PPk)

def= δ,k. The Pauli operators

form a basis for Hermitian operators acting on the system Hilbert space. In this basis, the lab state andthe MPS estimate can be represented by their characteristic functions

ρklab = tr

(Pkρlab

), σk = tr

(Pk∣∣∣ψ3

c

⟩ ⟨ψ3

c

∣∣∣) =⟨ψ3

c

∣∣∣ Pk∣∣∣ψ3

c

⟩(54)

respectively. The fidelity (53) is expressed in terms of the characteristic functions as

F(∣∣∣ψ3

c

⟩, ρlab

)=

4N∑k=1

ρklabσ

k. (55)

Estimating the fidelity by a straightforward application of Equation (55) requires measuring 4N observ-ables, each of which requires exponentially many (in N) measurements, and is thus infeasible. Thisimplies that over two hundred million observables need to be measured in our setting of fourteen spins,which is clearly infeasible.

The DFE method leverages the knowledge of the MPS estimate to overcome this infeasibility. Specif-ically, the full summation of Equation (55) is replaced by a preferential summation over those values ofk for which MPS-estimate components σk are likely to be large. In other words, more measurements aremade in those basis elements Pk for which the MPS estimate is known to have a large expectation value.

The preferential summation to obtain the fidelity estimate is performed as follows. First, the fidelity isexpressed as the expectation value

F(∣∣∣ψ3

c

⟩, ρ)=

4N∑k=1

ρklabσ

k =

4N∑k=1

qk ρklab

σk , (56)






24

of a random variable ρklab/σ

k over probability distributionqk def= (σk)2 : k = 1, 2, . . . , 4N

. Next,

this expectation value is estimated using a Monte Carlo approach. That is, M random indicesk1, k2, . . . , kM; km ∈ 1, 2, . . . , 4N

are generated according to the probability distribution qk where M

is chosen based on a desired threshold of error. In the experiment we set M = 250. In other words, thenumber of observables required to estimate the fidelity are reduced by six orders of magnitude.

The fidelity is obtained from the estimator

F def=

1M

M∑i=1

ρkilab

σki≈ F(∣∣∣ψ3

c

⟩, ρlab

), (57)

where ρkmlab estimates the lab-state expectation value (54). These estimates are obtained from measuring

M = 250 observables using many copies of the state for each observable, where a total of 5 × 105 copiesof the state were used. The number of copies Nk spent to measure a particular Pauli operator Pk waschosen to be proportional to the inverse square of its calculated expectation value σki for ψ3

c in order toprevent the error in the estimator F to be dominated by those terms of the sum in Eq. 57 for which σki isvery small.

The fidelity estimate (57) requires sampling the indices k1, k2, . . . , kM; this sampling is efficientlyperformed using classical algorithms outlined in the supplementary material of Reference [18]. Finally,the values σk1 , σk2 , . . . , σkM are determined by efficient MPS-based classical algorithms. This completesa summary of the direct fidelity estimation procedure.

The values of ρkmlab obtained in the experiment and the corresponding calculated σkm are depicted in

FIG. S10(a), for the M = 250 different observables, which are indexed by m. The distribution of ρkmlab/σ

km

for different m is presented in FIG. S10(b). Based on this distribution, we infer a fidelity estimate of 0.74.We present the procedure for calculating the error bars on this estimate in the next section.

B. Mean-square error and bias of DFE estimates

The fidelity estimate (57) is amenable to random error, which arises from (i) choosing a smaller num-ber M of indices than the maximum possible 4N and from (ii) using a finite number of measurementsto estimate the expectation values ρk1

lab, ρk2lab, . . . , ρ

kMlab. This random error is quantified by the variance

estimator

var[F] def=

1M(M − 1)

M∑m=1

ρkm

lab

σkm− F

2

, (58)

where F is determined using Equation 57. In the remainder of this section, we justify thatF(∣∣∣ψ3

c

⟩, ρlab

)(57) and var[F] (58) are unbiased estimators of the fidelity and the variance of the fidelity.

In other words, the expectation value of the random variables F and var[F] are respectively equal to thetrue value of the fidelity and the fidelity variance.

In order to account for random error from the experiment, we connect the fidelity estimator (57) andvariance estimator (58) with the measurement outcomes from the experiment. We account for randomerror in estimates ρkm

lab by expressing ρkmlab in terms of measurement outcomes:

ρklab = tr

(Pkρlab

)=

2N∑i=1

λki tr(∣∣∣ψk

i

⟩ ⟨ψk

i

∣∣∣ ρlab

)=

2N∑i=1

λki

⟨ψk

i

∣∣∣ ρlab∣∣∣ψk

i

⟩=

2N∑i=1

λki pk

i , (59)

where λki : k = 1, 2, . . . , 2N are the eigenvalues of Pauli operators

Pk =

2N∑i=1

λki

∣∣∣ψki

⟩ ⟨ψk

i

∣∣∣ , λki = ±

1√

2N, (60)

andpk

idef=⟨ψk

i

∣∣∣ ρlab∣∣∣ψk

i

⟩: k = 1, 2, . . . , 2N

is a probability distribution. Finally, the fidelity can be ex-

pressed as the expectation value

F(∣∣∣ψ3

c

⟩, ρlab

)=

4N∑k=1

qk ρklab

σk =

4N∑k=1

qk

∑2N

i=1 λki pk

i

σk

=

4N∑k=1

2N∑i=1

qk pki

λki

σk =

4N∑k=1

2N∑i=1

uki

λki

σk , (61)






25

of the random variable λkiσk over the probability distribution

uk

idef= qk pk

i : k = 1, 2, . . . , 4N , i = 1, 2, . . . , 2N.

In the experiment, we choose M observables P1, P2, . . . , PM. Each observable Pm is measured Nm

times, with each measurement returning outcome value λkmin

. The returned measurement outcomes areused to obtain expectation values as

ρkmlab =

1Nm

Nm∑n=1

λkmin. (62)

Thus, the fidelity estimate (57) is obtained from measurement outcomes as

F(∣∣∣ψ3

c

⟩, ρlab

)=

1M

M∑m=1

1Nm

∑Nmn=1 λ

kmin

σkm. (63)

The variance of F(∣∣∣ψ3

c

⟩, ρlab

)is obtained using estimator (58), which we express in terms of measurement

outcomes by the following simplification. Consider

var[F] def=

1M(M − 1)

M∑m=1

ρkm

lab

σkm− F

2

, (64)

=1

M(M − 1)

M∑m=1

(ρkm

lab

)2(σkm)2 − 2

ρkmlab

σkmF +(F)2 , (65)

=1

M(M − 1)

M∑m=1

(ρkm

lab

)2(σkm)2 −

MM(M − 1)

(F)2, (66)

where we have used the definition (57) to obtain (66) from (65). Substituting the expression for estimatorsF and ρk

lab, we obtain

var[F] =1

M(M − 1)

M∑m=1

(1

Nm

∑Nmn=1 λ

kmin

)2(σkm)2

− MM(M − 1)

1M

M∑m=1

1Nm

∑Nmn=1 λ

kmin

σkm

2

(67)

=1

M(M − 1)

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

− 1M2(M − 1)

M∑m,m′=1

1NmNm′

Nm∑n,n′=1

λkmin

σkm

λkm′in′

σkm′(68)

=1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

− 1M2(M − 1)

M∑m,m′=1mm′

1NmNm′

Nm∑n,n′=1

λkmin

σkm

λkm′in′

σkm′, (69)

which is the variance estimator in terms of measurement outcomesNow we show that the fidelity estimator (63) is an unbiased estimator. Consider the expectation value

of the fidelity

E[F] =E

1M

M∑m=1

1Nm

∑Nmn=1 λ

kmin

σkm

(70)

=1M

M∑m=1

1Nm

Nm∑n=1

E

λkm

in

σkm

. (71)

We note that the expectation value ofλkm

inσkm is equal to the true fidelity

E

λkm

in

σkm

=4N∑

km=

2N∑in=1

ukmin

λkmin

σkm= F(∣∣∣ψ3

c

⟩, ρlab

)(72)






26

because each of the λkmin

values are drawn from the same distribution for each value of m and n. Substitut-ing Equation (72) in the fidelity expectation value (71), we obtain

E[F] = F(∣∣∣ψ3

c

⟩, ρ), (73)

which implies that F is an unbiased estimator of the fidelity.Finally, we show var[F] is an unbiased estimator, i.e., that the expectation value of var[F] is the same

as the true variance

var[F] def= E

[F

2]−(E[F])2

(74)

of the estimator F. From Equation (69), we have the expectation value,

E(var[F]

)=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

− 1M2(M − 1)

M∑m,m′=1mm′

1NmNm′

Nm∑n,n′=1

λkmin

σkm

λkm′in′

σkm′

, (75)

which we simplify as

E(var[F]

)=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

− E

1

M2(M − 1)

M∑m,m′=1mm′

1NmNm′

Nm∑n,n′=1

λkmin

σkm

λkm′in′

σkm′

(76)

=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

− 1M2(M − 1)

M∑m,m′=1mm′

E

Nm∑n=1

1Nm

λkmin

σkm

E

Nm∑n′=1

1Nm′

λkm′in′

σkm′

(77)

=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

−1

M2(M − 1)

M∑m,m′=1mm′

(E[F])2, (78)

where in the last step we have used Equation (63) and that each λkmin

is drawn from the same distribution.

We add and subtract(E[F])2

to obtain

E(var[F]

)=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

−

1M2(M − 1)

M∑m,m′=1mm′

(E[F])2+(E[F])2 −

(E[F])2

(79)

=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

−1

M2(M − 1)

M∑m,m′=1mm′

(E[F])2

+1

M(M − 1)

M∑m,m′=1mm′

(E[F])2 −

(E[F])2

(80)

=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

+1

M2

M∑m,m′=1mm′

(E[F])2 −

(E[F])2. (81)






27

Performing the simplification of Equation (77)–(78), we combine the summations of the first two termsas

E(var[F]

)=E

1

M2

M∑m=1

1N2

m

Nm∑n,n′=1

λkmin

σkm

λkmin′

σkm

+1

M2

M∑m,m′=1mm′

E

Nm∑n=1

1Nm

λkmin

σkm

E

Nm∑n′=1

1Nm′

λkm′in′

σkm′

−(E[F])2

(82)

=1

M2

M∑m,m′=1

E

Nm∑n=1

1Nm

λkmin

σkm

Nm′∑n′=1

1Nm′

λkm′in′

σkm′

−(E[F])2

(83)

=E

[F

2]−(E[F])2, (84)

which is the same as the variance (74) of the fidelity estimate. Thus, we conclude that the varianceestimator var[F] is an unbiased estimator.

In summary, we have presented a procedure for estimating error bars on the DFE and have shown thatthe procedure returns an unbiased estimator of the variance. Using this procedure and the DFE proceduredescribed above we obtain the estimate 0.74 ± 0.05 for the fidelity of the experimental 14-ion state att14 = 4 ms with our MPS estimate.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

-3 -2 -1 0 1 2 3 4 50

5

10

15

20

25

Freq

uenc

y

(a) (b)

FIG. S10: Measured expectation values used for direct fidelity estimation of the quenched 14-spin MPS state.Outcomes are shown for measurements of the 14-spin quench state |ψ3

k〉 after simulator evolution time of t = 4 ms.For explanation of direct fidelity estimation (DFE) and definition of algebra used below, see Methods. (a) Scatterplot of the expected (for MPS state) and observed (for lab state) expectation values, σkm and ρkm

lab respectively, for thechosen M = 250 observables. Positive correlation is apparent. (b) The distribution of the random variable ρkm

lab/σkm

for the different observables. The mean (vertical red dashed line) and standard deviation of this distribution arethe respective estimators of the fidelity estimate and its error. For our experiment, the obtained fidelity estimate is0.74 ± 0.05.

VII. CERTIFIED MPS TOMOGRAPHY IS EFFICIENT FOR 1D LOCAL QUENCH DYNAMICS

A. Summary of the results

In this section, we show that certified MPS tomography can be used to characterise local quenchdynamics with resources that scale efficiently in system size. Specifically, we provide upper bounds tothe resources, both experimental and computational, required to characterise a state obtained by evolvinga pure product state under a 1D nearest-neighbour Hamiltonian [24]. These required resources grow nofaster than polynomially in the size N of the system, inverse polynomially with the tolerated infidelity Iof characterisation, and exponentially in the time t of evolution. That is, at any given time t during thequench dynamics, the resources to characterise the state scale efficiently (polynomially) in system size.

To show that certified MPS tomography is efficient for quenched states, a necessary condition is thatthese states admit an efficient (in N) MPS representation. This follows from simple arguments in additionto Corollary 3 of Reference [19]. However, the existence of the MPS is not sufficient to guarantee the






28

existence of a parent Hamiltonian with suitable spectral properties, which is essential in our certificationprocedure. In this section, we show that such a Hamiltonian for quenched states indeed exists, and thisexistence enables the use of the certified MPS tomography procedure for these states.

Our argument is structured as follows. First, we show that pure product states have parent Hamiltoniansthat have a unit gap above a non-degenerate ground state and that comprise local terms acting on singlesites only. Next, we generalise the pure product state result to quenched states, i.e., states that start outas pure product states and undergo a time evolution under a nearest-neighbour Hamiltonian. We showthat such states too can be well approximated by states which have a gapped parent Hamiltonian. Theseparent Hamiltonians comprise local terms that act on subsystems whose sizes 2Ω scale linearly in timeand logarithmically in N and in 1/I.

Physically, the existence of gapped parent Hamiltonians implies that the quenched states can beuniquely identified using only their local reductions because these Hamiltonians are local and have aunique ground state. Furthermore, by showing the existence of 2Ω-sized gapped parent Hamiltonians,we impose upper bounds on the required resources (number of measurements and computational time)required to characterise the state. Characterising a ground state with 2Ω-sized parent Hamiltonian re-quires measuring and classically processing a linear (in N) number of 2Ω-sized local reductions on a1D chain. This characterisation task requires resources (number of measurements and classical post-processing time) that scale exponentially in Ω and linearly in N via certified MPS tomography [6, 7].From this, and the scaling of Ω in the parameters N, I and t, we obtain the mentioned scaling of the ex-perimental and computational cost in terms of these parameters. In the next subsection, we recall resultsregarding gapped parent Hamiltonians of pure quantum states.

B. Background: parent Hamiltonian certificates

Here we recall briefly relevant notation regarding the parent Hamiltonian of a pure quantum state of Nqubits on a linear chain as introduced in Section IV A 3. The parent Hamiltonian of a pure state |ψ〉 is anyHermitian linear operator H such that |ψ〉 is the ground state of H. We assume that |ψ〉 is normalized, weset the ground state energy, i.e., the lowest eigenvalue 〈ψ|H |ψ〉, of H to zero and we denote by E1 ≥ 0the second smallest eigenvalue in which eigenvalues are repeated according to their multiplicities. TheHamiltonian is said to have a gap of size E1. The gap is non-zero (i.e., E1 > 0) if and only if the groundstate is non-degenerate.

Now we consider the energy of any arbitrary, possibly mixed, state ρ with respect to H and H isassumed to have a non-degenerate ground state. Then, the fidelity F(|ψ〉 , ρ) = 〈ψ| ρ |ψ〉 of ρ with respectto |ψ〉 is bounded below according to Eq. (9) with E0 = 0. That is,

F(|ψ〉 , ρ) ≥ 1 − tr(ρH)E1. (85)

Thus, the energy of ρ in terms of H provides a lower bound to the fidelity between |ψ〉 and ρ; we call alower bound to the fidelity a certificate.

The certification of the lab state using H is efficient, that is, requires number of measurements andcomputation-time that scale polynomially in the number of qudits if H is of the following form. Supposethat H is a sum

H =N−k+1∑

i=1

hi (86)

of terms which act non-trivially

hi = 11 ⊗ 12 ⊗ · · · ⊗ 1i−1 ⊗ hi,i+1,...,i+k−1 ⊗ 1i+k ⊗ · · · ⊗ 1N (87)

only on small (i.e., size k ∈ O(log N)) subsets of the complete system. Then, only the matching localreductions of ρ are necessary to obtain the energy tr(ρH):

tr(ρH) =∑

i

tr(ρhi) =∑

i

tr(ρihi), (88)

where ρi are the reduced density matrices that act on subsystem i, i + 1, . . . , i + k − 1. In this case thecertificate can be obtained from a number of measurements that scales linearly in the number of particles(and exponentially in the subsystem size k). This is an exponential improvement over the number ofmeasurements required for estimating fidelity by performing standard quantum state tomography. Fur-thermore, the summation of Eq. (88) can be performed in linear (in N) computational time as comparedto the exponentially large computation time required if the output from full tomography is used to ob-tain fidelity. In summary, determining the fidelity certificate is efficient with respect to measurement andcomputation time.






29

C. Product states have simple parent Hamiltonians

In this section, we show that pure product states admit a parent Hamiltonian that has unit gap and onlysingle-site local terms (Lemma 2). This result is a simple special case of prior work involving matrixproduct states [6, 10, 11].

Lemma 2. Let |ϕ〉 = |ϕ1〉 ⊗ |ϕ2〉 ⊗ · · · ⊗ |ϕN〉 be a product state on N qudits of dimension di ≥ 2,i ∈ 1, . . . ,N. Let 〈ϕi|ϕi〉 = 1 for all site indices i. Define

H =N∑

i=1

hi, hi = 11,...,i−1 ⊗ Pker(ρi) ⊗ 1i+1,...,N (89)

where Pker(ρi) is the orthogonal projection onto the null space of the reduced density operator ρi = |ϕi〉〈ϕi|of |ϕ〉 on site i. Then, the eigenvalues of H are given by 0, 1, 2, . . . ,N, the smallest eigenvalue zero isnon-degenerate and |ϕ〉 is an eigenvector of eigenvalue zero.

Proof. Expand the null space projectors hi in terms of the single-site pure states |ϕi〉 as

hi = 11,...,i−1 ⊗ Pker(ρi) ⊗ 1i+1,...,N

= 11,...,i−1 ⊗ (1i − |ϕi〉〈ϕi|) ⊗ 1i+1,...,N . (90)

We obtain a complete eigendecomposition of H as follows. For each site i, construct an orthonormalbasis |µi,1〉 , . . . , |µi,di〉 such that |µi,1〉 = |ϕi〉. Consider the corresponding orthonormal product basis of thecomplete system given by

|µL〉 = |µ1,1〉 ⊗ |µ2,2〉 ⊗ · · · ⊗ |µN,N 〉 , (91a)L = (1, 2, . . . , N), i ∈ 1, . . . , di. (91b)

We have

hi |µL〉 =

0, if i = 1,1, if i > 1.

(92)

and thus

H |µL〉 = λL |µL〉 , λL =∣∣∣∣i : i > 1, i ∈ 1, . . . ,N

∣∣∣∣. (93)

The basis vectors |µL〉 are seen to constitute an orthonormal eigenbasis of H and each eigenvalue λL isgiven by the number of tensor product factors |µi,i〉 with i > 1, i.e., with 〈ϕi|µi,i〉 = 0. Because we haverequired di ≥ 2, this already shows everything stated in the theorem: The eigenvalues of H are givenby 0, 1, 2, . . . ,N, the smallest eigenvalue zero is non-degenerate and |ϕ〉 is an eigenvector of eigenvaluezero.

D. Parent Hamiltonian for a locally time evolved state

This section presents the main results regarding the parent Hamiltonians of quenched states. We showthat the experimental and computational cost of characterising quenched states scales no faster than poly-nomially in N/I and exponentially in the quench time t.

Our proof is in two parts. First, we follow [20] to construct a state which closely approximates ourtime-evolved state. This approximate state is obtained by starting with a tensor product of pure states onat most Ω neighbouring sites and applying a single unitary operation that is a tensor product of unitarieson at most Ω sites (FIG. S11). Here, Ω depends on t, N and I but is seen to scale moderately.

Next, in Theorem 4 we show that this approximate state is the non-degenerate ground state of a suitableparent Hamiltonian. From Lemma 2, we know that the pure product state has a parent Hamiltonian withterms acting on at most Ω sites. The unitary operation does not increase the range of the terms in thepure-state parent Hamiltonian from Ω to more than 2Ω, which is small, i.e., O(log N), for suitably lowquench time O(log N). This parent Hamiltonian enables the usual certification procedure described inSec. VII B and the main text.

We construct the approximate state using the following theorem from Reference [20].

Theorem 3 (ε-QCA decomposition, [20]). Let H be a nearest-neighbour Hamiltonian on N qubits ina linear chain, i.e., H =

∑N−1i=1 hi,i+1. We fix a positive integer Ω and partition the linear chain into






30

N = 2N/Ω contiguous blocks each containing at most Ω/2 qubits (FIG. S11). There is an approximationof the time evolution operator U = e−iHt of the form

U′ =[U12 ⊗ U34 ⊗ · · · ⊗ UN−1,N

][V1 ⊗ V23 ⊗ V45 ⊗ · · · ⊗ VN−2,N−1 ⊗ VN

](94)

where U j, j+1, V j, j+1 and Vj are unitaries acting on the blocks specified by the subscripts. This approxi-mation satisfies ‖U − U′‖ ≤ ε under the restriction that

Ω ≥ c0|t| + c1 log(N/ε) (95)

where ‖·‖ is the operator norm and c0 and c1 are constants.Let |ψ(0)〉 = |ψ1〉 ⊗ · · · ⊗ |ψN〉 be an initial product state [25]. The state |ψ′〉 = U′ |ψ(0)〉 is an

approximation of the time-evolved state |ψ(t)〉 = U |ψ(0)〉 with ‖|ψ′〉 − |ψ(t)〉‖ ≤ ε. The approximation|ψ′〉 has a matrix product state representation with bond dimension no more than 2Ω.

Thus, the state |ψ′〉 closely approximates our time-evolved state. Specifically, if Ω is required to scalewith c′ log(N) where c′ is a suitable constant, then the norm-distance between |ψ〉 and |ψ′〉 scales as asinverse polynomial in N. Now we prove the existence of the parent Hamiltonian of |ψ′〉 and find an upperbound to the number of sites that the parent Hamiltonian terms act on.

FIG. S11: The ε-QCA decomposition of a unitary time evolution under a 1D nearest-neighbour Hamiltonian. Alinear chain of N spins is divided into N = 2N/Ω blocks, such that each block contains at most Ω/2 spins. Thedecomposition and the relation between N, Ω, approximation error ε and evolution time t have been introducedin Ref. [20]. We use the decomposition in Supplementary Material Sec. VII D to prove efficient certification ofMPS tomography for 1D local quench dynamics starting from a pure product state. This is accomplished with aparent Hamiltonian G whose local terms act on at most four blocks, i.e., at most 2Ω spins (Theorems 4 and 7 inSupplementary Material Sec. VII D).

Theorem 4. For |ψ′〉 as described in Theorem 3, there is a Hermitian linear operator G =∑N/Ω+1

j=1 g j

with non-degenerate smallest eigenvalue zero and eigenvector |ψ′〉, second smallest eigenvalue one andlargest eigenvalue N/Ω + 1. Each local term gj acts on no more than 2Ω consecutive qubits.

Proof. We define the intermediate product state

|φ〉 = [V1 ⊗ V23 ⊗ · · · ⊗ VN ] |ψ(0)〉 (96)=: |φ01〉 ⊗ |φ23〉 ⊗ · · · ⊗ |φN ,N+1〉 , (97)

i.e., |φ j, j+1〉 is a state on blocks j, j + 1. Blocks 0 and N + 1 are empty and have been introduced fornotational convenience [26]. From Lemma 2, we have a parent Hamiltonian

F = f01 + f23 + · · · + fN ,N+1 (98)

of |φ〉 with

f j, j+1 = 11,..., j−1 ⊗ Pker(|φ j, j+1〉〈φ j, j+1 |) ⊗ 1 j+2,...,N . (99)

Furthermore, F has a non-degenerate smallest eigenvalue zero with eigenvector |φ〉, second smallesteigenvalue one and largest eigenvalue N/2 + 1 = N/Ω + 1.

We define

U = U12 ⊗ U34 ⊗ · · · ⊗ UN−1,N (100)

and we define |ψ′〉 = U |φ〉. Because U is unitary, the operator G = UFU† has the same eigenvaluespectrum as F. That is, G has a non-degenerate smallest eigenvalue zero with eigenvector |ψ′〉, a secondsmallest eigenvalue one and a largest eigenvalue N/Ω + 1. We obtain the following representation of G

G = g12 + g1234 + g3456 + · · · + gN−1,N , (101)






31

where the identity operators are omitted and

g jklm = [U jk ⊗ Ulm] fkl [U jk ⊗ Ulm]†. (102)

The border terms are given by g12 = U12 f01U†12 and gN−1,N = UN−1,N fN ,N+1U†N−1,N . There are N/2 + 1terms in the sum and each term acts on at most four blocks, i.e., at most 2Ω qubits.

This completes our proof regarding the parent Hamiltonian of the approximate time-evolved state. Inthe following corollary, we use the parent Hamiltonian to obtain a fidelity certificate (Section VII B) forthe lab state ρ with respect to the approximate state.

Corrolary 5. Consider |ψ′〉 and G as described in Theorem 4 and define ψ′ def= |ψ′〉〈ψ′|. Denote by ‖·‖1

the trace norm of an operator. Let ρ be an arbitrary quantum state and let δ = ‖ρ − ψ′‖1. Then

〈ψ′| ρ |ψ′〉 ≥ 1 − tr(ρG) ≥ 1 − (N/Ω + 1)δ. (103)

Proof. As G has unit gap the fidelity lower bound (cf. Eq. (85)) becomes

〈ψ′| ρ |ψ′〉 ≥ 1 − tr(ρG)/1= 1 − tr(ρG), (104)

which is the first inequality of (103). Consider the energy tr(ρG) of ρ with respect to G. Using tr(ψ′G) =0, we have

tr(ρG) =∣∣∣tr(ρG) − tr(ψ′G)

∣∣∣ =∣∣∣tr([ρ − ψ′]G)

∣∣∣ . (105)

Thus, the energy

tr(ρG) ≤ ‖ρ − ψ′‖1‖G‖∞= (N/Ω + 1)δ (106)

is at most tr(ρG) ≤ (N/Ω+ 1)δ, where ‖·‖ = ‖·‖∞ denotes the operator norm. Combining Equations (104)and (106), we obtain the required inequalities.

The final Theorem requires the following Lemma:

Lemma 6. Let ‖·‖1 be the trace norm, ψ = |ψ〉〈ψ| and ψ′ = |ψ′〉〈ψ′|. If ‖|ψ〉 − |ψ′〉‖ ≤ ε ≤√

2, then‖ψ − ψ′‖1 ≤ 2ε.

Proof. Assume that ‖|ψ〉 − |ψ′〉‖ ≤ ε holds. This gives us

ε2 ≥ 2(1 −(〈ψ|ψ′〉)) ≥ 2(1 −√

F) (107)

where F = | 〈ψ|ψ′〉 |2 = F(|ψ〉 , |ψ′〉). This gives√

F ≥ 1 − ε2/2 and 1 − F = ε2 − ε4/4 ≤ ε2. The equality‖ψ − ψ′‖1 = 2

√1 − F completes the proof [12, Eqs. 9.11, 9.60, 9.99].

Our final theorem considers a state ρ close to a locally time-evolved state |ψ(t)〉; as before, |ψ′〉 is anapproximation of |ψ(t)〉. The theorem states the conditions under which the fidelity 〈ψ′| ρ |ψ′〉 can belower bounded by at least 1 − I, for some infidelity I:

Theorem 7. Let H be a nearest-neighbour Hamiltonian on N qubits in a linear chain, i.e., H =∑N−1i=1 hi,i+1. Let |ψ(0)〉 = |ψ1〉 ⊗ . . . |ψN〉 be an initial product state [27] and let |ψ(t)〉 = e−iHt |ψ(0)〉

be the time-evolved state. Define ψ(t) = |ψ(t)〉〈ψ(t)|.Let γ = ‖ρ − ψ(t)‖1 be the trace distance between an unknown state ρ and the time-evolved state. Fix

an infidelity I that satisfies I > 2Nγ. Choose an integer Ω ≥ 1 such that

Ω ≥ c0|t| + c1 log(

2NI/2N − γ

), (108)

with c0, c1 from Theorem 3. ψ′ = |ψ′〉〈ψ′| is the approximation of |ψ(t)〉 from the same theorem. We alsouse the parent Hamiltonian G from Theorem 4.

Then, the fidelity lower bound between |ψ′〉 and ρ will be at least

〈ψ′| ρ |ψ′〉 ≥ 1 − tr(ρG) ≥ 1 − I (109)






32

Proof. Theorem 3 applies for

ε =12

(I

2N− γ)

(110)

and it guarantees ‖|ψ(t)〉 − |ψ′〉‖ ≤ ε. As a consequence, we have ‖ψ(t) − ψ′‖1 ≤ 2ε (Lemma 6) and

‖ρ − ψ′‖1 ≤ ‖ρ − ψ(t)‖1 + ‖ψ(t) − ψ′‖1 ≤ γ + 2ε. (111)

In addition, we observe (using N ≥ 1 and Ω ≥ 1)

I = 2N(2ε + γ) ≥ (N + 1)(2ε + γ) ≥(NΩ+ 1)

(2ε + γ). (112)

We use Corollary 5 with δ ≤ 2ε + γ. It provides

〈ψ′| ρ |ψ′〉 ≥ 1 − tr(ρG) ≥ 1 − (N/Ω + 1)(2ε + γ) ≥ 1 − I, (113)

which is the required result.

If the experimental state ρ is the same as the quenched state |ψ(t)〉, then the requirement (108) for Ωchanges to

Ω ≥ c0|t| + c2 log(N) + c3 log(

1I

)+ c4 (114)

which enables us to quantify the resources required for certification.

E. Conclusion

In summary, the time-evolved state |ψ(t)〉 from Theorem 7 can be certified up to infidelity I withrespect to a state |ψ′〉, which has a parent Hamiltonian G with non-degenerate ground state and unitgap. The local terms in G act on at most 2Ω sites. Ω can be chosen as the lowest integer that satisfiesEquation (114) depending on the evolution time t, number of qubits N and infidelity I. Note thatΩ growsno faster than linearly with time and logarithmically with N/I.

The existence of gapped parent Hamiltonians with 2Ω-sized terms means that the quenched statescan be uniquely identified, in principle, using only 2Ω-sized local reductions. Whether such a state canactually be obtained using existing numerical algorithms is discussed in References [6, 7]. Although, noformal proofs for the convergence of these algorithms are available, we observe (main text) that thesealgorithms perform well in practice. Theorem 7 complements these discussions by ensuring that thefidelity of any reconstructed state with respect to the lab state can always be bounded from below. If thislower bound is smaller than desired, then the numerical algorithms can be run again, perhaps with moremeasurements to reduce quantum projection noise or with measurements on larger-sized subsystems tocapture all correlations.

Theorem 7 also imposes upper bounds on the required number of measurements and the requiredcomputational time for characterising the state. Specifically, the experimental and computational costsfor performing certified MPS tomography of quenched states scale no faster than polynomially in N,inverse polynomially in I and exponentially in the quench time t. Thus, certified MPS tomography isefficient in the size of the system and in the inverse infidelity tolerance for quenched states.






33

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Cambridge, 2007), 9 edn.[13] Jones, E., Oliphant, T., Peterson, P. et al. SciPy: Open source scientific tools for Python (2001). URL http:

//www.scipy.org/.[14] Suess, D. & Holzäpfel, M. mpnum: A matrix product representation library for python (2016). URL https:

//github.com/dseuss/mpnum.[15] Jaynes, E. T. Probability Theory: The Logic of Science (Cambridge University Press, Cambridge, 2003).[16] Bennett, C. H., Bernstein, H. J., Popescu, S. & Schumacher, B. Concentrating partial entanglement by local

operations. Phys. Rev. A 53, 2046–2052 (1996).[17] Flammia, S. T. & Liu, Y.-K. Direct Fidelity Estimation from Few Pauli Measurements. Phys. Rev. Lett. 106,

230501 (2011).[18] da Silva, M. P., Landon-Cardinal, O. & Poulin, D. Practical Characterization of Quantum Devices without

Tomography. Phys. Rev. Lett. 107, 210404 (2011).[19] Brandão, F. G. S. L. & Horodecki, M. Exponential decay of correlations implies area law. Commun. Math.

Phys. 333, 761–798 (2014).[20] Osborne, T. J. Efficient approximation of the dynamics of one-dimensional quantum spin systems. Phys. Rev.

Lett. 97, 157202 (2006). quant-ph/0508031.[21] One can also make a direct connection from the POVM description of the 6k outcome probabilities to the ability

to reconstruct the corresponding (local k-spin reduced) density matrix. See the paragraph following Eq. (36) onpage 12.

[22] Whenever there is some freedom in choosing the csi, jl′ , we choose them with equal magnitude.[23] A similar problem, where probabilities are a quadratic instead of a linear function of a pure density matrix ρ,

has been discussed in the supplementary material of Ref. [18].[24] The results in this section hold for nearest-neighbour Hamiltonians. Strictly speaking, the interaction in the lab

is not of this kind but similar results are expected to hold because the lab interaction is finite ranged.[25] One could also allow a product state on the N blocks instead of product states on N qubits.[26] For the application of Lemma 2 in the next sentence, the empty blocks 0 and N + 1 shall be dropped.[27] One could also allow a product state on the N blocks instead of product states on N qubits.






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